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OJEE Integrals — practice questions

Explore 56 free Integrals questions in OJEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Integrals questions and exercises with answers

Showing questions 1–30 of 56.

Find $\displaystyle\int x^4\,dx$.Read the answer and worked solution →Find $\displaystyle\int \sin x\,dx$.Read the answer and worked solution →Evaluate $\displaystyle\int_{0}^{\pi/2} \sin x\,dx$.Read the answer and worked solution →Find $\displaystyle\int 2x \cos(x^2)\,dx$.Read the answer and worked solution →Find $\displaystyle\int x e^x\,dx$.Read the answer and worked solution →Find $\displaystyle\int \dfrac{1}{(x-1)(x+1)}\,dx$.Read the answer and worked solution →∫ x dx equals:Read the answer and worked solution →∫ cos x dx equals:Read the answer and worked solution →∫ sin x dx equals:Read the answer and worked solution →∫ eˣ dx equals:Read the answer and worked solution →∫ dx/x for x > 0 equals:Read the answer and worked solution →∫ sec² x dx equals:Read the answer and worked solution →∫ (3x² - 4x + 5) dx equals:Read the answer and worked solution →Definite integral ∫₀¹ x² dx equals:Read the answer and worked solution →∫ sec x tan x dx equals:Read the answer and worked solution →By substitution u = x², ∫ 2x sin(x²) dx equals:Read the answer and worked solution →∫ dx/(1 + x²) equals:Read the answer and worked solution →∫ dx/√(1 - x²) equals:Read the answer and worked solution →∫ x eˣ dx (using integration by parts):Read the answer and worked solution →∫₀^(π/2) sin x dx equals:Read the answer and worked solution →∫ ln x dx (by parts):Read the answer and worked solution →∫₀¹ x e^(x²) dx equals:Read the answer and worked solution →∫ dx/(x ln x) equals:Read the answer and worked solution →∫ (x² + 1)/(x² - 1) dx equals:Read the answer and worked solution →∫ sin³ x dx (using sin³ x = sin x (1 - cos² x)):Read the answer and worked solution →∫ dx/√(x² + a²) equals:Read the answer and worked solution →∫ dx/(x² + 4) equals:Read the answer and worked solution →∫₀^(π) sin² x dx equals:Read the answer and worked solution →∫₋ₐ^a x³ dx equals (for any a):Read the answer and worked solution →∫ cosec² x dx equals:Read the answer and worked solution →

Worked practice questions

Find $\displaystyle\int x^4\,dx$.

  1. $5x^5 + C$
  2. $\dfrac{x^5}{5} + C$
  3. $x^5 + C$
  4. $4x^3 + C$
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Correct answer: B. $\dfrac{x^5}{5} + C$

Power rule for integration: $\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \neq -1$. Plug $n = 4$: $\dfrac{x^5}{5} + C$. Option A is the *derivative* of $x^4$, not the integral. Always check by differentiating your answer.

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Find $\displaystyle\int \sin x\,dx$.

  1. $\cos x + C$
  2. $-\cos x + C$
  3. $-\sin x + C$
  4. $\sin x + C$
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Correct answer: B. $-\cos x + C$

$\dfrac{d}{dx}(-\cos x) = \sin x$, so $\displaystyle\int \sin x\,dx = -\cos x + C$. Mnemonic: differentiating gives a negative on the sine term ($\dfrac{d}{dx} \cos x = -\sin x$). To reverse, you keep the sign that makes the derivative *positive* $\sin x$, which is the $-\cos x$ choice.

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Evaluate $\displaystyle\int_{0}^{\pi/2} \sin x\,dx$.

  1. $-1$
  2. $1$
  3. $\pi/2$
  4. $0$
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Correct answer: B. $1$

$\displaystyle\int_{0}^{\pi/2} \sin x\,dx = [-\cos x]_{0}^{\pi/2} = -\cos(\pi/2) + \cos(0) = -0 + 1 = 1$. Geometric interpretation: this is the area under one quarter of the sine curve from $0$ to $\pi/2$, which equals exactly $1$ in standard units.

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