JEE Mathematics Definite Integration MCQs Set 23

Practice MCQs for JEE Mathematics Definite Integration

Explore reliable objective questions for Definite Integration tailored for JEE learners. Utilizing these Mathematics multiple-choice formats ensures thorough preparation and strengthens problem-solving speed for upcoming school assessments.

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View or download the dedicated Definite Integration MCQ resource below. Practicing these 50 objective questions regularly builds familiarity with standard exam patterns and helps secure higher marks in final Mathematics evaluations.

Multiple Choice Questions

Question 1: Evaluate \( \int_{\pi/4}^{\pi/3}\operatorname{cosec}x\,d(\sin x) \) =
(a) \( \frac12\log\frac32 \)
(b) \( \log2 \)
(c) \( 0 \)
(d) \( \log\frac32 \)
Show Answer & Explanation

Answer: (a) \( \frac12\log\frac32 \)

Explanation:
1. cosec x d(sin x) = cosec x cos x dx = cot x dx.
2. ∫ cot x dx = log sin x, from π/4 to π/3.
3. log(√3/2) - log(1/√2) = log(√3/√2) = (1/2) log(3/2).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 2: Evaluate \( \int_{-\pi/2}^{\pi/2}\sin^7x\,dx \) =
(a) \( 2 \)
(b) \( 0 \)
(c) \( 3 \)
(d) \( 1 \)
Show Answer & Explanation

Answer: (b) \( 0 \)

Explanation:
1. sin⁷x is an odd function.
2. The integral over a symmetric interval is 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 3: \( \int\limits_{0}^{\frac{\pi}{4}} \tan^{4}{\left(x \right)}\, dx = \)
(a) \( \frac{2}{3} + \frac{\pi}{4} \)
(b) \( \frac{3}{2} + \frac{\pi^{2}}{4} \)
(c) \( \frac{\pi}{4} \)
(d) \( - \frac{2}{3} + \frac{\pi}{4} \)
Show Answer & Explanation

Answer: (d) \( - \frac{2}{3} + \frac{\pi}{4} \)

Explanation:
1. tan⁴x = tan²x sec²x - tan²x = tan²x sec²x - sec²x + 1.
2. ∫0^(π/4) = [tan³x/3 - tan x + x] = 1/3 - 1 + π/4 = π/4 - 2/3.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 4: \( \int\limits_{0}^{\frac{\pi}{4}} \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = \)
(a) \( \frac{7 \pi}{512} \)
(b) \( \frac{3 \pi}{512} \)
(c) \( - \frac{\pi^{2}}{512} \)
(d) \( \frac{\pi}{512} \)
Show Answer & Explanation

Answer: (b) \( \frac{3 \pi}{512} \)

Explanation:
1. sin x cos x = (1/2) sin 2x, so the integrand is (1/16) sin⁴2x.
2. ∫0^(π/4) = (1/32)∫0^(π/2) sin⁴t dt = (1/32)(3π/16) = 3π/512.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 5: \( \int\limits_{0}^{\frac{\pi}{2}} \frac{1}{5 \cos{\left(x \right)} + 4}\, dx = \)
(a) \( \frac{\log{\left(2 \right)}}{2} \)
(b) \( \frac{\log{\left(2 \right)}}{5} \)
(c) \( \frac{\log{\left(3 \right)}}{3} \)
(d) \( \frac{\log{\left(2 \right)}}{3} \)
Show Answer & Explanation

Answer: (d) \( \frac{\log{\left(2 \right)}}{3} \)

Explanation:
1. Put tan(x/2) = t: 4 + 5cos x = (9 - t²)/(1 + t²) and dx = 2dt/(1 + t²).
2. ∫0^1 2dt/(9 - t²) = (1/3) log((3 + t)/(3 - t)) from 0 to 1 = (1/3) log 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 6: If \( I_{(m,n)}=\int_0^1t^m(1+t)^ndt \), then \( I_{(m,n)} \) in terms of \( I_{(m+1,n-1)} \) is
(a) \( \frac{2^n}{m+1}+\frac n{m+1}I_{(m+1,n-1)} \)
(b) \( \frac{2^n}{m+1}-\frac n{m+1}I_{(m+1,n-1)} \)
(c) \( \frac n{m+1}I_{(m+1,n-1)} \)
(d) \( \frac m{m+1}I_{(m+1,n-1)} \)
Show Answer & Explanation

Answer: (b) \( \frac{2^n}{m+1}-\frac n{m+1}I_{(m+1,n-1)} \)

Explanation:
1. Integrate by parts with u = (1 + t)^n and dv = t^m dt.
2. I = [(1 + t)^n t^(m+1)/(m + 1)] from 0 to 1 - (n/(m + 1))∫t^(m+1)(1 + t)^(n-1) dt.
3. = 2^n/(m + 1) - (n/(m + 1)) I_(m+1, n-1).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 7: \( \int\limits_{- \frac{\pi}{4}}^{\frac{\pi}{4}} \log{\left(\sin{\left(x \right)} + \cos{\left(x \right)} \right)}\, dx = \)
(a) \( - \frac{\pi \log{\left(2 \right)}}{4} \)
(b) \( \frac{\pi^{2} \log{\left(2 \right)}}{4} \)
(c) \( - \pi \log{\left(2 \right)} \)
(d) \( \pi \log{\left(2 \right)} \)
Show Answer & Explanation

Answer: (a) \( - \frac{\pi \log{\left(2 \right)}}{4} \)

Explanation:
1. The integral is ∫0^(π/4)[log(cos x + sin x) + log(cos x - sin x)] dx = ∫0^(π/4) log cos 2x dx.
2. Put 2x = t: (1/2)∫0^(π/2) log cos t dt = (1/2)(-(π/2) log 2) = -(π/4) log 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 8: If \( f(x)=\begin{cases}\sin x,&0\le x<\pi/2\\1,&\pi/2\le x<3\\e^{x-3},&3\le x\le9\end{cases} \), then \( \int_0^9f(x)\,dx \) =
(a) \( 3+\frac\pi2+e^6 \)
(b) \( 3-\frac\pi2+e^6 \)
(c) \( 3-\frac\pi2-e^6 \)
(d) \( 3+\frac\pi2-e^6 \)
Show Answer & Explanation

Answer: (b) \( 3-\frac\pi2+e^6 \)

Explanation:
1. ∫0^(π/2) sin x dx = 1.
2. ∫_(π/2)^3 1 dx = 3 - π/2.
3. ∫3^9 e^(x - 3) dx = e⁶ - 1.
4. Total = 1 + 3 - π/2 + e⁶ - 1 = 3 - π/2 + e⁶.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 9: For positive \( a \) and \( b \), \( \int_0^\infty\frac{dx}{(x^2+a^2)(x^2+b^2)} \) =
(a) \( \frac{\pi}{2(a+b)} \)
(b) \( \frac{\pi(a+b)}{ab} \)
(c) \( \frac{\pi ab}{a+b} \)
(d) \( \frac{\pi}{2ab(a+b)} \)
Show Answer & Explanation

Answer: (d) \( \frac{\pi}{2ab(a+b)} \)

Explanation:
1. 1/((x² + a²)(x² + b²)) = [1/(b² - a²)][1/(x² + a²) - 1/(x² + b²)].
2. ∫0^∞ = [1/(b² - a²)](π/(2a) - π/(2b)) = π(b - a)/(2ab(b² - a²)).
3. = π/(2ab(a + b)).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 10: If \( \int_e^xt\,f(t)\,dt=\sin x-x\cos x-\frac{x^2}2 \) for all \( x\in\mathbb R\setminus\{0\} \), then \( f(\pi/6) \) =
(a) \( 1 \)
(b) \( -\frac12 \)
(c) \( 0 \)
(d) \( \frac32 \)
Show Answer & Explanation

Answer: (b) \( -\frac12 \)

Explanation:
1. Differentiate both sides: x f(x) = cos x - cos x + x sin x - x = x sin x - x.
2. So f(x) = sin x - 1.
3. f(π/6) = 1/2 - 1 = -1/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 11: \( \int\limits_{0}^{\frac{\pi}{2}} \log{\left(\cot{\left(x \right)} \right)}\, dx = \)
(a) \( 1 \)
(b) \( 0 \)
(c) \( \pi \)
(d) \( \frac{\pi}{2} \)
Show Answer & Explanation

Answer: (b) \( 0 \)

Explanation:
1. Replace x by π/2 - x: ∫0^(π/2) log cot x dx = ∫0^(π/2) log tan x dx.
2. Adding, 2I = ∫0^(π/2) log(1) dx = 0.
3. So I = 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 12: Let \( f:\mathbb R\to\mathbb R \) be a continuous function with \( f(x+y)=f(x)+f(y) \) for all \( x,y \). If \( \int_0^2f(x)dx=\alpha \), then \( \int_{-2}^2f(x)dx \) is equal to
(a) \( \alpha-2 \)
(b) \( 2\alpha \)
(c) \( \alpha \)
(d) \( 0 \)
Show Answer & Explanation

Answer: (d) \( 0 \)

Explanation:
1. A continuous additive function is f(x) = λx, which is odd.
2. The integral of an odd function over [-2, 2] is 0.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 13: If \( a<0(a) \( 0 \)
(b) \( a+b \)
(c) \( b-a \)
(d) \( a-b \)
Show Answer & Explanation

Answer: (b) \( a+b \)

Explanation:
1. Split at 0: |x|/x = -1 on (a, 0) and 1 on (0, b).
2. ∫ = -(0 - a) + (b - 0) = a + b.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 14: The integral \( \int_0^\pi xf(\sin x)\,dx \) equals
(a) \( \pi\int_0^{\pi/2}f(\cos x)dx \)
(b) \( \pi\int_0^{\pi}f(\cos x)dx \)
(c) \( \pi\int_0^{\pi}f(\sin x)dx \)
(d) \( 2\pi \)
Show Answer & Explanation

Answer: (a) \( \pi\int_0^{\pi/2}f(\cos x)dx \)

Explanation:
1. Replace x by π - x: I = ∫0^π (π - x) f(sin x) dx, so I = (π/2)∫0^π f(sin x) dx.
2. Since f(sin x) is symmetric about π/2, this is π∫0^(π/2) f(sin x) dx.
3. Replacing x by π/2 - x gives π∫0^(π/2) f(cos x) dx.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 15: \( \int\limits_{0}^{\pi} \left(\cos{\left(x \right)} + \left|{\cos{\left(x \right)}}\right|\right)\, dx = \)
(a) \( 2 \)
(b) \( \frac{1}{2} \)
(c) \( 1 \)
(d) \( -1 \)
Show Answer & Explanation

Answer: (a) \( 2 \)

Explanation:
1. √((1 + cos 2x)/2) = |cos x|.
2. ∫0^π cos x dx = 0 and ∫0^π |cos x| dx = 2.
3. Total = 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 16: Evaluate \( \int_0^\pi[\cot x]\,dx \) (where [ ] is the greatest integer function) =
(a) \( -1 \)
(b) \( 1 \)
(c) \( -\frac\pi2 \)
(d) \( \frac\pi2 \)
Show Answer & Explanation

Answer: (c) \( -\frac\pi2 \)

Explanation:
1. Replace x by π - x: [cot(π - x)] = [-cot x].
2. For non-integer cot x, [cot x] + [-cot x] = -1, so 2I = ∫0^π (-1) dx = -π.
3. I = -π/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 17: If \( f(x)=\int_0^x\left(\sin^4t+\cos^4t\right)dt \), then \( f(x+\pi) \) is equal to
(a) \( f(x)+f\left(\frac\pi2\right) \)
(b) \( f(x)-2f\left(\frac\pi2\right) \)
(c) \( f(x)+2f\left(\frac\pi2\right) \)
(d) \( f(x)-f\left(\frac\pi2\right) \)
Show Answer & Explanation

Answer: (c) \( f(x)+2f\left(\frac\pi2\right) \)

Explanation:
1. sin⁴t + cos⁴t has period π/2.
2. f(x + π) = f(x) + ∫_x^(x+π) (sin⁴t + cos⁴t) dt = f(x) + ∫0^π (sin⁴t + cos⁴t) dt (period π).
3. ∫0^π = 2∫0^(π/2) = 2f(π/2). So f(x + π) = f(x) + 2f(π/2).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 18: Evaluate \( \lim_{n\to\infty}\left\{\left(1+\frac1n\right)\left(1+\frac2n\right)\cdots\left(1+\frac nn\right)\right\}^{1/n} \) =
(a) \( \frac8e \)
(b) \( \frac4e \)
(c) \( \frac1e \)
(d) \( e \)
Show Answer & Explanation

Answer: (b) \( \frac4e \)

Explanation:
1. Taking logs, log y = lim (1/n)Σ log(1 + r/n) = ∫0^1 log(1 + x) dx.
2. ∫0^1 log(1 + x) dx = 2 log 2 - 1.
3. y = e^(2 log 2 - 1) = 4/e.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 19: Evaluate \( \lim_{n\to\infty}\left(\frac{1^4}{1^5+n^5}+\frac{2^4}{2^5+n^5}+\cdots+\frac{n^4}{n^5+n^5}\right) \) =
(a) \( \frac15\log3 \)
(b) \( \frac15\log2 \)
(c) \( \frac14\log3 \)
(d) \( \frac14\log2 \)
Show Answer & Explanation

Answer: (b) \( \frac15\log2 \)

Explanation:
1. Divide numerator and denominator by n⁵: each term is (1/n)(r/n)⁴/((r/n)⁵ + 1).
2. The limit is ∫0^1 x⁴/(x⁵ + 1) dx = (1/5) log(x⁵ + 1) from 0 to 1 = (1/5) log 2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 20: If \( f(x)=\begin{vmatrix}\sin x+\sin2x+\sin3x&\sin2x&\sin3x\\3+4\sin x&3&4\sin x\\1+\sin x&\sin x&1\end{vmatrix} \), then \( \int_0^{\pi/2}f(x)\,dx \) is
(a) \( 0 \)
(b) \( 3 \)
(c) \( \frac23 \)
(d) \( \frac13 \)
Show Answer & Explanation

Answer: (d) \( \frac13 \)

Explanation:
1. Apply C₁ → C₁ - (C₂ + C₃): the first column becomes (sin x, 0, 0).
2. f(x) = sin x(3 - 4 sin²x) = sin 3x.
3. ∫0^(π/2) sin 3x dx = [-cos 3x/3] = 1/3.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 21: \( \int\limits_{0}^{9} \left(\left\lfloor{\sqrt{x}}\right\rfloor + 2\right)\, dx = \)
(a) \( 27 \)
(b) \( 31 \)
(c) \( 22 \)
(d) \( 23 \)
Show Answer & Explanation

Answer: (b) \( 31 \)

Explanation:
1. [√x + 2] = 2 + [√x].
2. ∫0^9 2 dx = 18; [√x] = 0 on [0, 1), 1 on [1, 4), 2 on [4, 9).
3. ∫0^9 [√x] dx = 1(4 - 1) + 2(9 - 4) = 13.
4. Total = 18 + 13 = 31.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 22: \( \int\limits_{0}^{\pi} \frac{x}{\cos^{2}{\left(x \right)} + 1}\, dx = \)
(a) \( 0 \)
(b) \( \frac{\sqrt{2} \pi^{2}}{4} \)
(c) \( \frac{\pi^{2}}{4} \)
(d) \( \frac{\pi^{2}}{8} \)
Show Answer & Explanation

Answer: (b) \( \frac{\sqrt{2} \pi^{2}}{4} \)

Explanation:
1. I = (π/2)∫0^π dx/(1 + cos²x) = π∫0^(π/2) dx/(1 + cos²x).
2. Divide by cos²x and put tan x = t: ∫0^∞ dt/(2 + t²) = π/(2√2).
3. I = π²/(2√2).

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

Question 23: Evaluate \( \int_a^b\frac{f(x)}{f(x)+f(a+b-x)}dx \) =
(a) \( a-b \)
(b) \( \frac{b-a}2 \)
(c) \( b-a \)
(d) \( \frac{a-b}2 \)
Show Answer & Explanation

Answer: (b) \( \frac{b-a}2 \)

Explanation:
1. Replace x by a + b - x and add the two forms.
2. 2I = ∫a^b dx = b - a.
3. So I = (b - a)/2.

Teacher's Note:
1. Look for a standard property of definite integrals such as symmetry, the king property or periodicity.
2. Split the interval where the integrand changes its form.

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