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JEE Main Integrals and Applications — practice questions — page 3

Explore 98 free Integrals and Applications questions in JEE Main Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Integrals and Applications questions and exercises with answers

Showing questions 61–90 of 98.

By Walli's formula, ∫_0^(π/2) sin^4(x) dx equals:Read the answer and worked solution →Compute ∫ 2x sin(x²) dx using substitution:Read the answer and worked solution →The integral ∫_{-1}^{1} x³ dx equals:Read the answer and worked solution →Complete the linearity rulesRead the answer and worked solution →Complete the relationship between the two calculiRead the answer and worked solution →True or false: the constant of integrationRead the answer and worked solution →Differentiation or integration?Read the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Flip and pair: term with meaningRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →True or false: reading the notationRead the answer and worked solution →Reading the definitions backRead the answer and worked solution →True or false: what the constant meansRead the answer and worked solution →Tap every true statement about integrationRead the answer and worked solution →To evaluate $\int x\cos x\, dx$, the best technique is:Read the answer and worked solution →By the Fundamental Theorem of Calculus, ∫_0^2 (3x² + 2x) dx equals:Read the answer and worked solution →Tap every property of indefinite integrals the chapter givesRead the answer and worked solution →The integral $\int x^3\, dx$ equals:Read the answer and worked solution →The AREA bounded by y = x² and the x-axis from x = 0 to x = 3 is:Read the answer and worked solution →The improper integral ∫_1^∞ (1/x²) dx evaluates to:Read the answer and worked solution →$\displaystyle\int x^n\,dx$ (where $n \ne -1$) equals:Read the answer and worked solution →The integral ∫ x^n dx (for n ≠ -1) equals:Read the answer and worked solution →Using integration by parts, $\displaystyle\int x e^x\,dx$ equals:Read the answer and worked solution →To solve the homogeneous differential equation $\dfrac{dy}{dx} = \dfrac{x^2 + y^2}{xy}$, the appropriate substitution is:Read the answer and worked solution →Using partial fractions, $\displaystyle\int \dfrac{1}{(x-1)(x-2)}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{x^2 - a^2}\,dx$ (with $|x| > a > 0$) equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{1+x^2}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \dfrac{1}{\sqrt{1 - x^2}}\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \sec^2 x\,dx$ equals:Read the answer and worked solution →$\displaystyle\int \log x\,dx$ equals:Read the answer and worked solution →

Worked practice questions

By Walli's formula, ∫_0^(π/2) sin^4(x) dx equals:

  1. $3\pi/16$ (using even n formula)
  2. $\pi/4$ (incorrect calculation)
  3. $\pi/8$ (using odd n formula)
  4. $\pi/2$ (full quarter)
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Correct answer: A. $3\pi/16$ (using even n formula)

Walli's for n = 4 (even): = (3/4)(1/2)(π/2) = 3π/16. For odd n: omit the π/2 factor.

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Compute ∫ 2x sin(x²) dx using substitution:

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

Let u = x², du = 2x dx → ∫sin(u) du = -cos(u) + C = -cos(x²) + C. Substitution exploits the chain rule in reverse.

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The integral ∫_{-1}^{1} x³ dx equals:

  1. $2/4 = 0.5$ (using power rule alone)
  2. $0$ (odd function, symmetric limits)
  3. $1$ (computing only the upper limit)
  4. $2$ (full range, doubled value)
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Correct answer: B. $0$ (odd function, symmetric limits)

x³ is an ODD function (f(-x) = -f(x)). For an odd function over symmetric limits [-a, a], the integral is 0. The areas above x-axis (0 to 1) and below (-1 to 0) cancel.

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