CBSE Class 12 Mathematics Integrals MCQs Set 02

Question. \( \int \frac{1}{x^{3/4}} dx \) is equal to
(a) \( x^{1/4} + C \)
(b) \( 4x^{1/4} + C \)
(c) \( \frac{x^{1/4}}{4} + C \)
(d) \( \frac{x^{3/4}}{4} + C \)
Answer: (b) \( 4x^{1/4} + C \)

Question. \( \int 5^x dx \) is equal to
(a) \( \log 5 + C \)
(b) \( \frac{5^x}{\log x} + C \)
(c) \( \frac{5^x}{\log 5} + C \)
(d) \( 5\log x + C \)
Answer: (c) \( \frac{5^x}{\log 5} + C \)

Question. \( \int e^{5\log x} dx \) is equal to
(a) \( \frac{x^5}{5} + C \)
(b) \( \frac{x^6}{6} + C \)
(c) \( 5x^4 + C \)
(d) \( 6x^5 + C \)
Answer: (b) \( \frac{x^6}{6} + C \)

Question. If \( f'(x) = x + \frac{1}{x} \), then \( f(x) \) is
(a) \( x^2 + \log|x| + C \)
(b) \( \frac{x^2}{2} + \log|x| + C \)
(c) \( \frac{x}{2} + \log|x| + C \)
(d) \( \frac{x^2}{2} - \log|x| + C \)
Answer: (b) \( \frac{x^2}{2} + \log|x| + C \)

Question. \( \int \frac{\sec x}{\sec x - \tan x} dx \) equals
(a) \( \sec x - \tan x + C \)
(b) \( \sec x + \tan x + C \)
(c) \( \tan x - \sec x + C \)
(d) \( -(\sec x + \tan x) + C \)
Answer: (b) \( \sec x + \tan x + C \)

Question. \( \int x^2 e^{x^3} dx \) is equal to
(a) \( \frac{1}{3}e^{x^3} + C \)
(b) \( \frac{1}{3}e^{x^2} + C \)
(c) \( \frac{1}{2}e^{x^3} + C \)
(d) \( \frac{1}{2}e^{x^2} + C \)
Answer: (a) \( \frac{1}{3}e^{x^3} + C \)

Question. \( \int \frac{1}{x(\log x)^2} dx \) is equal to
(a) \( 2\log(\log x) + C \)
(b) \( -\frac{1}{\log x} + C \)
(c) \( \frac{(\log x)^3}{3} + C \)
(d) \( \frac{3}{(\log x)^3} + C \)
Answer: (b) \( -\frac{1}{\log x} + C \)

Question. \( \int e^{3\log x} (x^4 + 1)^{-1} dx \) is equal to
(a) \( \frac{1}{4}\log(x^4 + 1) + C \)
(b) \( -\log\left(\frac{1}{x^4+1}\right) + C \)
(c) \( \frac{x^3}{x^4+1} + C \)
(d) \( \frac{\log|x^4+1|}{x^4+1} + C \)
Answer: (a) \( \frac{1}{4}\log(x^4 + 1) + C \)

Question. \( \int \tan x dx \) is equal to
(a) \( \sec x + C \)
(b) \( \sec^2 x + C \)
(c) \( \log|\sec x| + C \)
(d) \( \log|\cos x| + C \)
Answer: (c) \( \log|\sec x| + C \)

Question. The integral \( \int \frac{dx}{\sqrt{9-4x^2}} \) is equal to
(a) \( \frac{1}{6}\sin^{-1}\left(\frac{2x}{3}\right) + C \)
(b) \( \frac{1}{2}\sin^{-1}\left(\frac{2x}{3}\right) + C \)
(c) \( \sin^{-1}\left(\frac{2x}{3}\right) + C \)
(d) \( \frac{3}{2}\sin^{-1}\left(\frac{2x}{3}\right) + C \)
Answer: (b) \( \frac{1}{2}\sin^{-1}\left(\frac{2x}{3}\right) + C \)

Question. \( \int \frac{dx}{x^2 - 16} \) is equal to
(a) \( \frac{1}{4}\log\left|\frac{x-4}{x+4}\right| + C \)
(b) \( \frac{1}{8}\log\left|\frac{x-4}{x+4}\right| + C \)
(c) \( \frac{1}{16}\log\left|\frac{x-4}{x+4}\right| + C \)
(d) \( \frac{1}{2}\log\left|\frac{x-4}{x+4}\right| + C \)
Answer: (b) \( \frac{1}{8}\log\left|\frac{x-4}{x+4}\right| + C \)

Question. If \( \frac{d}{dx}f(x) = \log x \), then \( f(x) \) equals
(a) \( -\frac{1}{x} + C \)
(b) \( x(\log x - 1) + C \)
(c) \( x(\log x + x) + C \)
(d) \( \frac{1}{x} + C \)
Answer: (b) \( x(\log x - 1) + C \)

Question. Find integration of \( \tan^{-1} x \).
(a) \( x\tan^{-1} x - \frac{1}{2}\log(1+x^2) + C \)
(b) \( x\tan^{-1} x + \frac{1}{2}\log(1+x^2) + C \)
(c) \( \frac{x^2}{2}\tan^{-1} x - \frac{1}{2}\log(1+x^2) + C \)
(d) \( \frac{x^2}{2}\tan^{-1} x + \frac{1}{2}\log(1+x^2) + C \)
Answer: (a) \( x\tan^{-1} x - \frac{1}{2}\log(1+x^2) + C \)

Question. \( \int \frac{x-3}{(x-1)^3} e^x dx \) is equal to
(a) \( \frac{2e^x}{(x-1)^3} + C \)
(b) \( \frac{-2e^x}{x-1} + C \)
(c) \( \frac{e^x}{x-1} + C \)
(d) \( \frac{e^x}{(x-1)^2} + C \)
Answer: (d) \( \frac{e^x}{(x-1)^2} + C \)

Question. If \( \int_0^a 3x^2 dx = 8 \), then the value of \( a \) is
(a) 2
(b) 4
(c) 8
(d) 10
Answer: (a) 2

Question. \( \int_{-1}^1 \frac{|x-2|}{x-2} dx \), \( x \neq 2 \) is equal to
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (d) -2

Question. For any integer \( n \), the value of \( \int_0^\pi e^{\sin^2 x} \cos^3(2n+1)x dx \) is
(a) -1
(b) 0
(c) 1
(d) 2
Answer: (b) 0

Question. \( \int_0^{\pi/6} \sec^2\left(x - \frac{\pi}{6}\right) dx \) is equal to
(a) \( \frac{1}{\sqrt{3}} \)
(b) \( -\frac{1}{\sqrt{3}} \)
(c) \( \sqrt{3} \)
(d) \( -\sqrt{3} \)
Answer: (a) \( \frac{1}{\sqrt{3}} \)

Question. Find value of \( \int_{\pi/4}^{\pi/2} \cot \theta \csc^2 \theta d\theta \).
(a) \( \frac{1}{2} \)
(b) \( -\frac{1}{2} \)
(c) 0
(d) \( -\frac{\pi}{8} \)
Answer: (a) \( \frac{1}{2} \)

Question. \( \int_0^4 (e^{2x} + x) dx \) is equal to
(a) \( \frac{15 + e^8}{2} \)
(b) \( \frac{16 - e^8}{2} \)
(c) \( \frac{e^8 - 15}{2} \)
(d) \( \frac{-e^8 - 15}{2} \)
Answer: (a) \( \frac{15 + e^8}{2} \)

Question. The value of \( \int_{-1}^1 x|x| dx \) is
(a) \( \frac{1}{6} \)
(b) \( \frac{1}{3} \)
(c) \( -\frac{1}{6} \)
(d) 0
Answer: (d) 0

Question. Which of the expression is equal to \( \int_0^{\pi/2} \log(\tan x) dx \)?
(a) \( \int_0^{\pi/2} \log(\sin x) dx \)
(b) \( \int_0^{\pi/2} \log(\cos x) dx \)
(c) \( \int_0^{\pi/2} \log(\sec x) dx \)
(d) \( \int_0^{\pi/2} \log(\cot x) dx \)
Answer: (d) \( \int_0^{\pi/2} \log(\cot x) dx \)

Question. \( \int_0^1 \log\left(\frac{1-x}{x}\right) dx \) is
(a) \( \log 3 - \log 2 \)
(b) \( \log 2 - \log 3 \)
(c) \( \log 4 - \log 5 \)
(d) 0
Answer: (d) 0

Question. If \( \int_{-2}^3 x^2 dx = k \int_{-2}^0 x^2 dx + \int_0^3 x^2 dx \), then the value of \( k \) is
(a) 2
(b) 1
(c) 0
(d) \( \frac{1}{2} \)
Answer: (a) 2

Question. The value of \( \int_1^e \log x dx \) is
(a) 0
(b) 1
(c) \( e \)
(d) \( e \log e \)
Answer: (b) 1

Assertion-Reason Based Questions

Directions In the questions given below are two statements labelled as Assertion (A) and Reason (R). In the context of the two statements, which one of the following is correct?
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.

Question. Assertion (A) \( \int \frac{1}{x^{5/3}} dx \) is equal to \( -\frac{3}{2}x^{-2/3} + C \).
Reason (R) \( \int x^n dx \) gives \( \frac{x^{n+1}}{n+1} + C \), \( n \neq -1 \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int (4-3x)^7 dx \) is equal to \( -\frac{1}{24}(4-3x)^8 + C \).
Reason (R) \( \int (ax+b)^n dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C \), \( n \neq -1 \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int \frac{1}{\sqrt{1+\cos 2x}} dx = \frac{1}{\sqrt{2}}\log|\sec x + \tan x| + C \).
Reason (R) \( \int \cot x dx = \log|\sin x| + C \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (b) Both A and R are correct; R is not the correct explanation of A.

Question. Assertion (A) \( \int \tan^3 x \sec^3 x dx = \frac{1}{5}\sec^5 x - \frac{1}{3}\sec^3 x + C \).
Reason (R) \( \int x^n dx \) gives \( \frac{x^{n+1}}{n+1} + C \), \( n \neq -1 \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int \frac{1}{9+4x^2} dx = \frac{1}{6}\tan^{-1}\left(\frac{2x}{3}\right) + C \).
Reason (R) \( \int \frac{1}{x^2+a^2} dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int \frac{1}{\sqrt{4-x^2}} dx = \sin^{-1}\left(\frac{x}{2}\right) + C \).
Reason (R) \( \int \frac{1}{\sqrt{a^2-x^2}} dx = \sin^{-1}\left(\frac{x}{a}\right) + C \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int x \log x dx = \frac{1}{2}x^2 \log x - \frac{x^2}{4} + C \).
Reason (R) \( \int u \cdot v dx = u \int v dx - \int \left( \frac{d}{dx}(u) \int v dx \right) dx \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) The value of \( \int_0^1 \frac{dx}{1+x^2} \) is \( \frac{3\pi}{4} \).
Reason (R) \( \int_a^b f(x) dx \) is known as definite integral and is given by \( \int_a^b f(x) dx = g(b) - g(a) \), \( g(x) \) is anti-derivative of \( f(x) \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (d) R is correct; A is incorrect.

Question. Assertion (A) \( \int_{-\pi}^\pi (1-x^2)\sin x \cos^2 x dx = 0 \).
Reason (R) \( \int_{-a}^a f(x) dx = 0 \), if \( f(x) \) is an odd function.
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Question. Assertion (A) \( \int_2^8 \frac{\sqrt{10-x}}{\sqrt{x}+\sqrt{10-x}} dx = 3 \).
Reason (R) \( \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \).
(a) Both A and R are correct; R is the correct explanation of A.
(b) Both A and R are correct; R is not the correct explanation of A.
(c) A is correct; R is incorrect.
(d) R is correct; A is incorrect.
Answer: (a) Both A and R are correct; R is the correct explanation of A.

Case Study Based Questions - I

The derivative of the velocity function \( v(t) \) with respect to time \( t \), gives us the acceleration of the object. This can be written as \( \frac{dv}{dt} = a(t) \) (in \( \text{m/s}^2 \)).
The derivative of the displacement function \( x(t) \) with respect to time \( t \), gives us the velocity of the object. This can be written as \( \frac{dx}{dt} = v(t) \) (in \( \text{m/s} \)).
A car is moving at a constant velocity of \( 15 \text{ m/s} \). It starts decelerating when it is about to reach its destination with acceleration, \( a(t) = -\frac{t}{3}\text{ m/s}^2 \), where \( t \) is the time (in seconds) for which it decelerated before coming to rest.
Based on the above information, answer the following questions.

Question. Find the velocity function of the car after it started decelerating. Show your steps.
Answer: Given acceleration \( a(t) = -\frac{t}{3} \).
We know that velocity function \( v(t) = \int a(t) dt \).
\( \Rightarrow v(t) = \int -\frac{t}{3} dt = -\frac{t^2}{6} + C \).
At \( t = 0 \), \( v(0) = 15 \).
\( \Rightarrow 15 = -\frac{0}{6} + C \Rightarrow C = 15 \).
On putting the value of \( C \), we get:
\( v(t) = -\frac{t^2}{6} + 15 \).

Question. Find the time it takes the car to stop after it starts decelerating. Show your work and give a valid reason.
Answer: The car stops when its velocity becomes zero, i.e., \( v(t) = 0 \).
From the velocity function, we have:
\( -\frac{t^2}{6} + 15 = 0 \)
\( \Rightarrow -\frac{t^2}{6} = -15 \Rightarrow t^2 = 90 \)
\( \Rightarrow t = \pm \sqrt{90} = \pm 3\sqrt{10} \text{ s} \).
Since time cannot be negative, \( t = 3\sqrt{10} \text{ s} \).

Question. Find the displacement function of the car and calculate its displacement from the moment it starts decelerating till it stops. Show your steps.
Answer: The displacement function \( x(t) \) is given by:
\( x(t) = \int v(t) dt = \int \left(15 - \frac{t^2}{6}\right) dt = 15t - \frac{t^3}{18} + C' \).
At \( t = 0 \), \( x(0) = 0 \Rightarrow C' = 0 \).
Thus, \( x(t) = 15t - \frac{t^3}{18} \).
The displacement till it stops (at \( t = 3\sqrt{10} \)) is:
\( x(3\sqrt{10}) = 15(3\sqrt{10}) - \frac{(3\sqrt{10})^3}{18} = 45\sqrt{10} - \frac{270\sqrt{10}}{18} = 45\sqrt{10} - 15\sqrt{10} = 30\sqrt{10} \text{ m} \).

Case Study Based Questions - II

We know that a given function \( f(x) \) is said to be an even function, if \( f(-x) = f(x) \) and an odd function when \( f(-x) = -f(x) \), also
\( \int_{-a}^a f(x)dx = \begin{cases} 2 \int_0^a f(x)dx, & \text{if } f(x) \text{ is even} \\ 0, & \text{if } f(x) \text{ is odd} \end{cases} \)
Based on the above information, answer the following questions.

Question. Find whether \( f(x) = x^3 \sin^{40} x \) is even or odd function.
Answer: Let \( f(x) = x^3 \sin^{40} x \).
\( f(-x) = (-x)^3 \cdot \sin^{40}(-x) = -x^3 \cdot (-\sin x)^{40} = -x^3 \sin^{40} x = -f(x) \).
Hence, \( f(x) \) is an odd function.

Question. Find whether \( f(x) = \sin|x| + \cos|x| \) is even or odd function.
Answer: Let \( f(x) = \sin|x| + \cos|x| \).
\( f(-x) = \sin|-x| + \cos|-x| = \sin|x| + \cos|x| = f(x) \).
Hence, \( f(x) \) is an even function.

Question. Find the value of \( \int_{-\pi}^\pi x^{17} \cos^4 x dx \).
Answer: Let \( f(x) = x^{17} \cos^4 x \).
\( f(-x) = (-x)^{17} \cos^4(-x) = -x^{17} \cos^4 x = -f(x) \).
Since \( f(x) \) is an odd function, we have \( \int_{-\pi}^\pi x^{17} \cos^4 x dx = 0 \).

Question. Find the value of \( \int_{-2}^2 (x^3 + x\cos x + \tan^5 x + 1) dx \).
Answer: Let \( I = \int_{-2}^2 (x^3 + x\cos x + \tan^5 x + 1) dx \).
We can split the integral as:
\( I = \int_{-2}^2 x^3 dx + \int_{-2}^2 x\cos x dx + \int_{-2}^2 \tan^5 x dx + \int_{-2}^2 1 dx \).
Here, \( x^3 \), \( x\cos x \), and \( \tan^5 x \) are odd functions, so their integrals over \( [-2, 2] \) are all 0.
The constant function \( 1 \) is even. Thus:
\( I = 0 + 0 + 0 + 2\int_0^2 1 dx = 2[x]_0^2 = 4 \).

Case Study Based Questions - III

Let \( f(x) \) be a function and \( f'(x) \) be its derivative. Now, teacher asks Raju, "can you answer my question." Raju says, "What is your question?" Teacher wrote question on blackboard:
\( \int e^x [f(x) + f'(x)] dx = ? \)
After reading the question, Raju gave the answer as \( e^x f(x) + C \).
On the basis of above information, answer the following questions.

Question. If \( \int e^x \left(\frac{1}{x^2} - \frac{2}{x^3}\right) dx = e^x f(x) + C \), then find the value of \( f(x) \).
Answer: Let \( f(x) = \frac{1}{x^2} \). Then \( f'(x) = -\frac{2}{x^3} \).
Using the formula \( \int e^x [f(x) + f'(x)] dx = e^x f(x) + C \), we get:
\( \int e^x \left(\frac{1}{x^2} - \frac{2}{x^3}\right) dx = e^x \left(\frac{1}{x^2}\right) + C \).
On comparing, we get \( f(x) = \frac{1}{x^2} \).

Question. Find the value of \( \int e^x \sec x(1 + \tan x) dx \).
Answer: Let \( I = \int e^x (\sec x + \sec x \tan x) dx \).
Let \( f(x) = \sec x \). Then \( f'(x) = \sec x \tan x \).
Using the formula \( \int e^x [f(x) + f'(x)] dx = e^x f(x) + C \), we get:
\( I = e^x \sec x + C \).

Question. Find the value of \( \int \frac{e^x}{x} \{x(\log x)^2 + 2\log x\} dx \).
Answer: Let \( I = \int e^x \left\{(\log x)^2 + \frac{2\log x}{x}\right\} dx \).
Let \( f(x) = (\log x)^2 \). Then \( f'(x) = \frac{2\log x}{x} \).
Thus, \( I = e^x (\log x)^2 + C \).

Question. Find the value of \( \int [\tan(\log x) + \sec^2(\log x)] dx \).
Answer: Let \( I = \int [\tan(\log x) + \sec^2(\log x)] dx \).
Put \( \log x = t \Rightarrow x = e^t \Rightarrow dx = e^t dt \).
\( \Rightarrow I = \int (\tan t + \sec^2 t) e^t dt \).
Here, \( f(t) = \tan t \) and \( f'(t) = \sec^2 t \).
Thus, \( I = e^t \tan t + C = x \tan(\log x) + C \).

Case Study Based Questions - IV

We have two properties of definite integration. First is \( \int_0^a f(x) dx = \int_0^a f(a - x) dx \) and second is \( \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \).
Based on the above information, answer the following questions.

Question. Find the value of \( \int_0^{\pi/2} \sin^2 x dx \).
Answer: Let \( I = \int_0^{\pi/2} \sin^2 x dx \) ... (i)
Using the property \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \):
\( I = \int_0^{\pi/2} \sin^2\left(\frac{\pi}{2} - x\right) dx = \int_0^{\pi/2} \cos^2 x dx \) ... (ii)
Adding (i) and (ii):
\( 2I = \int_0^{\pi/2} (\sin^2 x + \cos^2 x) dx = \int_0^{\pi/2} 1 dx = [x]_0^{\pi/2} = \frac{\pi}{2} \).
\( \Rightarrow I = \frac{\pi}{4} \).

Question. Find the value of \( \int_0^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} dx \).
Answer: Let \( I = \int_0^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} dx \) ... (i)
Using the property \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \):
\( I = \int_0^{\pi/2} \frac{\sin(\pi/2-x) - \cos(\pi/2-x)}{1 + \sin(\pi/2-x)\cos(\pi/2-x)} dx = \int_0^{\pi/2} \frac{\cos x - \sin x}{1 + \cos x \sin x} dx \) ... (ii)
Adding (i) and (ii):
\( 2I = \int_0^{\pi/2} \frac{(\sin x - \cos x) + (\cos x - \sin x)}{1 + \sin x \cos x} dx = \int_0^{\pi/2} 0 dx = 0 \).
\( \Rightarrow I = 0 \).

Question. Find the value of \( \int_0^{\pi/4} \log(1 + \tan x) dx \).
Answer: Let \( I = \int_0^{\pi/4} \log(1 + \tan x) dx \) ... (i)
Using the property \( \int_0^a f(x) dx = \int_0^a f(a-x) dx \):
\( I = \int_0^{\pi/4} \log\left(1 + \tan\left(\frac{\pi}{4} - x\right)\right) dx \).
Since \( \tan\left(\frac{\pi}{4}-x\right) = \frac{1 - \tan x}{1 + \tan x} \), we have:
\( I = \int_0^{\pi/4} \log\left(1 + \frac{1 - \tan x}{1 + \tan x}\right) dx = \int_0^{\pi/4} \log\left(\frac{2}{1 + \tan x}\right) dx \)
\( \Rightarrow I = \int_0^{\pi/4} [\log 2 - \log(1 + \tan x)] dx = \int_0^{\pi/4} \log 2 dx - I \)
\( \Rightarrow 2I = \log 2 [x]_0^{\pi/4} = \frac{\pi}{4} \log 2 \).
\( \Rightarrow I = \frac{\pi}{8} \log 2 \).

Question. Find the value of \( \int_{\pi/6}^{\pi/3} \log(\tan x) dx \).
Answer: Let \( I = \int_{\pi/6}^{\pi/3} \log(\tan x) dx \) ... (i)
Using the property \( \int_a^b f(x) dx = \int_a^b f(a+b-x) dx \):
Here, \( a+b = \frac{\pi}{6} + \frac{\pi}{3} = \frac{\pi}{2} \).
\( I = \int_{\pi/6}^{\pi/3} \log\left(\tan\left(\frac{\pi}{2} - x\right)\right) dx = \int_{\pi/6}^{\pi/3} \log(\cot x) dx \) ... (ii)
Adding (i) and (ii):
\( 2I = \int_{\pi/6}^{\pi/3} (\log(\tan x) + \log(\cot x)) dx = \int_{\pi/6}^{\pi/3} \log(\tan x \cot x) dx = \int_{\pi/6}^{\pi/3} \log 1 dx = 0 \).
\( \Rightarrow I = 0 \).