JEE Mathematics Definite Integration MCQs Set 27

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Multiple Choice Questions

Question 1: The value of \( I=\int_0^1x(1-x)^n\,dx \) is
(a) \( \frac1{n+1}-\frac1{n+2} \)
(b) \( \frac1{n+2} \)
(c) \( \frac1{n+1} \)
(d) \( \frac1{n+1}+\frac1{n+2} \)
Show Answer & Explanation

Answer: (a) \( \frac1{n+1}-\frac1{n+2} \)

Explanation:
1. Write x = 1 - (1 - x): x(1 - x)^n = (1 - x)^n - (1 - x)^(n+1).
2. ∫0^1 = 1/(n + 1) - 1/(n + 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: If \( f(x)=\begin{cases}\dfrac{|x|}x,&x\ne0\\1,&x=0\end{cases} \) on \( [-4,5] \), then \( \int_{-4}^5f(x)\,dx \) =
(a) \( -1 \)
(b) \( 0 \)
(c) \( 1 \)
(d) \( \frac12 \)
Show Answer & Explanation

Answer: (c) \( 1 \)

Explanation:
1. f(x) = -1 for x < 0 and 1 for x ≥ 0.
2. ∫_(-4)^0 (-1) dx = -4 and ∫0^5 1 dx = 5.
3. Total = 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 3: For \( n>2 \) (an integer), \( \int_0^{\pi/4}\left[\tan^n(x-[x])+\tan^{n-2}(x-[x])\right]dx \) =
(a) \( \frac1{n-2} \)
(b) \( \frac1{n+1} \)
(c) \( \frac1{n-1} \)
(d) \( \frac1n \)
Show Answer & Explanation

Answer: (c) \( \frac1{n-1} \)

Explanation:
1. On [0, π/4], [x] = 0, so the integrand is tan^n x + tan^(n-2) x = tan^(n-2) x sec²x.
2. ∫0^(π/4) tan^(n-2) x sec²x dx = 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 4: Evaluate \( \int_{-1}^{3}\left(\tan^{-1}\frac x{x^2+1}+\tan^{-1}\frac{x^2+1}x\right)dx \) =
(a) \( 2\pi \)
(b) \( 3\pi \)
(c) \( \pi \)
(d) \( 4\pi \)
Show Answer & Explanation

Answer: (c) \( \pi \)

Explanation:
1. For y > 0, tan⁻¹y + tan⁻¹(1/y) = π/2; for y < 0 it is -π/2.
2. The argument x/(x² + 1) has the sign of x.
3. ∫_(-1)^0 (-π/2) dx + ∫0^3 (π/2) dx = -π/2 + 3π/2 = π.

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: Evaluate \( \lim_{x\to0}\frac{\int_0^x\sin t\,dt}{\sin^2x} \) =
(a) \( 1 \)
(b) \( \frac12 \)
(c) \( -1 \)
(d) \( 2 \)
Show Answer & Explanation

Answer: (b) \( \frac12 \)

Explanation:
1. The numerator is 1 - cos x.
2. (1 - cos x)/sin²x = 1/(1 + cos x) → 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 6: \( \int\limits_{0}^{\frac{\pi}{2}} \frac{1}{4 \sin^{2}{\left(x \right)} + 1}\, dx = \)
(a) \( \frac{\sqrt{5} \pi}{5} \)
(b) \( \frac{\sqrt{5} \pi}{15} \)
(c) \( \frac{\sqrt{5} \pi}{10} \)
(d) \( \frac{\pi}{2} \)
Show Answer & Explanation

Answer: (c) \( \frac{\sqrt{5} \pi}{10} \)

Explanation:
1. Divide by cos²x: ∫ sec²x dx/(1 + 5tan²x).
2. Put tan x = t: ∫0^∞ dt/(1 + 5t²) = (1/√5)(π/2) = π/(2√5).

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: \( \int\limits_{0}^{\infty} e^{- 2 x} \sin{\left(5 x \right)}\, dx = \)
(a) \( \frac{2}{29} \)
(b) \( - \frac{2}{29} \)
(c) \( \frac{7}{25} \)
(d) \( \frac{5}{29} \)
Show Answer & Explanation

Answer: (d) \( \frac{5}{29} \)

Explanation:
1. Use ∫0^∞ e^(-ax) sin bx dx = b/(a² + b²).
2. With a = 2, b = 5: 5/29.

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: Evaluate \( \int_{-20\pi}^{20\pi}|\sin x|\,[\sin x]\,dx \) (where [ ] is the greatest integer function) =
(a) \( -30 \)
(b) \( -20 \)
(c) \( -40 \)
(d) \( -60 \)
Show Answer & Explanation

Answer: (c) \( -40 \)

Explanation:
1. Replace x by -x: I = ∫|sin x|[-sin x] dx.
2. Adding, 2I = ∫|sin x|([sin x] + [-sin x]) dx = -∫_(-20π)^(20π) |sin x| dx (as [t] + [-t] = -1 for non-integers).
3. ∫_(-20π)^(20π) |sin x| dx = 40 × 2 = 80, so I = -40.

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: \( \int\limits_{0}^{\pi} \frac{1}{2 \sin{\left(x \right)} + \cos{\left(x \right)} + 3}\, dx = \)
(a) \( \frac{\pi}{6} \)
(b) \( \frac{\pi}{4} \)
(c) \( \frac{\pi}{2} \)
(d) \( \frac{\pi}{3} \)
Show Answer & Explanation

Answer: (b) \( \frac{\pi}{4} \)

Explanation:
1. Put tan(x/2) = t: 3 + 2 sin x + cos x = (2t² + 4t + 4)/(1 + t²).
2. The integral is ∫0^∞ dt/(t² + 2t + 2) = [tan⁻¹(t + 1)] from 0 to ∞ = π/2 - π/4 = π/4.

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 \( [x] \) is the greatest integer function, \( \int_{0.5}^{4.5}[x]\,dx+\int_{-1}^{1}|x|\,dx \) =
(a) \( 6 \)
(b) \( 9 \)
(c) \( 8 \)
(d) \( 7 \)
Show Answer & Explanation

Answer: (b) \( 9 \)

Explanation:
1. On [0.5, 4.5], [x] = 0, 1, 2, 3, 4 on intervals of lengths 0.5, 1, 1, 1, 0.5, so ∫ = 0 + 1 + 2 + 3 + 2 = 8.
2. ∫_(-1)^1 |x| dx = 1.
3. Total = 9.

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: If \( I_n=\int_0^{\pi/4}\tan^nx\,dx \), then \( \lim_{n\to\infty}n\left[I_n+I_{n+2}\right] \) =
(a) \( 0 \)
(b) \( \infty \)
(c) \( 1 \)
(d) \( \frac12 \)
Show Answer & Explanation

Answer: (c) \( 1 \)

Explanation:
1. I_n + I_(n+2) = ∫0^(π/4) tan^n x sec²x dx = 1/(n + 1).
2. n/(n + 1) → 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 12: \( \int\limits_{0}^{2} \left|{x^{2} + 2 x - 3}\right|\, dx = \)
(a) \( 0 \)
(b) \( 2 \)
(c) \( 4 \)
(d) \( -1 \)
Show Answer & Explanation

Answer: (c) \( 4 \)

Explanation:
1. x² + 2x - 3 = (x + 3)(x - 1) is negative on (0, 1) and positive on (1, 2).
2. ∫0^1 -(x² + 2x - 3) dx = 5/3 and ∫1^2 (x² + 2x - 3) dx = 7/3.
3. Total = 4.

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: Evaluate \( \int_{-1}^{1}\frac{\sin x-x^2}{3-|x|}dx \) =
(a) \( 2\int_0^1\frac{\sin x-x^2}{3-|x|}dx \)
(b) \( 0 \)
(c) \( \int_0^1\frac{-2x^2}{3-|x|}dx \)
(d) \( \int_0^1\frac{\sin x}{3-|x|}dx \)
Show Answer & Explanation

Answer: (c) \( \int_0^1\frac{-2x^2}{3-|x|}dx \)

Explanation:
1. sin x/(3 - |x|) is odd, so its integral over [-1, 1] is 0.
2. -x²/(3 - |x|) is even, so ∫_(-1)^1 = 2∫0^1 -x²/(3 - |x|) dx = ∫0^1 -2x²/(3 - |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 14: Evaluate \( \lim_{n\to\infty}\left[\frac1{\sqrt{2n-1^2}}+\frac1{\sqrt{4n-2^2}}+\frac1{\sqrt{6n-3^2}}+\cdots+\frac1n\right] \) =
(a) \( \frac\pi3 \)
(b) \( \frac\pi4 \)
(c) \( \frac\pi2 \)
(d) \( \frac\pi6 \)
Show Answer & Explanation

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

Explanation:
1. The r-th term is 1/√(2rn - r²) = (1/n)·1/√(2(r/n) - (r/n)²), for r = 1 to n.
2. The limit is ∫0^1 dx/√(2x - x²) = ∫0^1 dx/√(1 - (x - 1)²) = [sin⁻¹(x - 1)] = 0 + π/2 = 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 15: If \( I=\int_0^1\sqrt{1+x^3}\,dx \), then
(a) \( I>\frac{\sqrt7}2 \)
(b) \( I>1 \)
(c) \( I<1 \)
(d) \( I>\frac{\sqrt5}2 \)
Show Answer & Explanation

Answer: (b) \( I>1 \)

Explanation:
1. For 0 < x < 1, 1 < √(1 + x³) < √2.
2. So ∫0^1 1 dx < I, i.e. I > 1.
3. (The true value is about 1.11, which is less than √5/2, about 1.118.)

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: \( \int\limits_{\frac{1}{3}}^{1} \frac{\sqrt[3]{- x^{3} + x}}{x^{4}}\, dx = \)
(a) \( 0 \)
(b) \( 6 \)
(c) \( 3 \)
(d) \( 4 \)
Show Answer & Explanation

Answer: (b) \( 6 \)

Explanation:
1. Write the integrand as (1/x² - 1)^(1/3)/x³.
2. Put 1/x² - 1 = t, so -2dx/x³ = dt; the limits become 8 to 0.
3. ∫ = (1/2)∫0^8 t^(1/3) dt = (1/2)(3/4)(16) = 6.

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: \( \int\limits_{\pi}^{10 \pi} \left|{\sin{\left(x \right)}}\right|\, dx = \)
(a) \( 10 \)
(b) \( 20 \)
(c) \( 8 \)
(d) \( 18 \)
Show Answer & Explanation

Answer: (d) \( 18 \)

Explanation:
1. |sin x| has period π and ∫0^π |sin x| dx = 2.
2. The interval [π, 10π] has 9 periods, giving 9 × 2 = 18.

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

Answer: (d) \( \frac{\log{\left(3 \right)}}{4} \)

Explanation:
1. Put sin x - cos x = u: du = (cos x + sin x) dx and 3 + sin 2x = 4 - u².
2. ∫_(-1)^0 du/(4 - u²) = (1/4) log((2 + u)/(2 - u)) from -1 to 0 = (1/4) log 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 19: \( \int\limits_{0}^{\frac{1}{2}} \frac{x \sin^{-1}{\left(x \right)}}{\sqrt{1 - x^{2}}}\, dx = \)
(a) \( \frac{\sqrt{3} \pi}{12} \)
(b) \( - \frac{\sqrt{3} \pi}{12} + \frac{1}{2} \)
(c) \( - \frac{\sqrt{3} \pi}{12} \)
(d) \( \frac{\sqrt{3} \pi}{12} + \frac{1}{2} \)
Show Answer & Explanation

Answer: (b) \( - \frac{\sqrt{3} \pi}{12} + \frac{1}{2} \)

Explanation:
1. By parts with u = sin⁻¹x and dv = x dx/√(1 - x²), v = -√(1 - x²).
2. I = [-√(1 - x²) sin⁻¹x] + ∫0^(1/2) dx = -(√3/2)(π/6) + 1/2.
3. = 1/2 - π√3/12.

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

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

Explanation:
1. Put x = tan θ: the integral is ∫0^(π/2) log sec²θ dθ = -2∫0^(π/2) log cos θ dθ.
2. = -2(-(π/2) log 2) = π 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 21: If \( \int_0^1\frac{e^t}{t+1}dt=a \), then \( \int_{b-1}^b\frac{e^{-t}dt}{t-b-1} \) is equal to
(a) \( ae^{-b} \)
(b) \( -ae^{-b} \)
(c) \( -be^{-a} \)
(d) \( ae^{b} \)
Show Answer & Explanation

Answer: (b) \( -ae^{-b} \)

Explanation:
1. Put t = b - y, so dt = -dy; the limits become 1 to 0.
2. The integral becomes ∫1^0 e^(-b+y)/(-y - 1)(-dy) = -e^(-b)∫0^1 e^y/(y + 1) dy.
3. = -a e^(-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 22: \( \int\limits_{- \frac{\pi}{2}}^{\frac{\pi}{2}} \sin{\left(\left|{x}\right| \right)}\, dx = \)
(a) \( 2 \)
(b) \( -1 \)
(c) \( \frac{1}{2} \)
(d) \( 1 \)
Show Answer & Explanation

Answer: (a) \( 2 \)

Explanation:
1. sin|x| is even.
2. ∫_(-π/2)^(π/2) = 2∫0^(π/2) sin x dx = 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 \( \lim_{x\to0}\frac{\left(\int_0^xe^{t^2}dt\right)^2}{\int_0^xe^{2t^2}dt} \) =
(a) \( 2 \)
(b) \( 0 \)
(c) \( 3 \)
(d) \( 1 \)
Show Answer & Explanation

Answer: (b) \( 0 \)

Explanation:
1. Both numerator and denominator tend to 0.
2. By L'Hospital's rule the ratio tends to 2e^(x²)(∫0^x e^(t²) dt)/e^(2x²).
3. As x → 0, ∫0^x e^(t²) dt → 0, so the limit 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 24: \( \int\limits_{0}^{\frac{\pi}{2}} \frac{200 \sin{\left(x \right)} + 100 \cos{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = \)
(a) \( 75 \pi \)
(b) \( 50 \pi \)
(c) \( 25 \pi \)
(d) \( 150 \pi \)
Show Answer & Explanation

Answer: (a) \( 75 \pi \)

Explanation:
1. Replace x by π/2 - x and add: 2I = 300∫0^(π/2) dx = 150π.
2. I = 75π.

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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