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JEE Advanced Unit 6B: Definite Integration & Riemann Sums — practice questions

Explore 42 free Unit 6B: Definite Integration & Riemann Sums questions in JEE Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 6B: Definite Integration & Riemann Sums questions and exercises with answers

Showing questions 1–30 of 42.

Evaluate $\displaystyle\lim_{n\to\infty} \sum_{r=1}^{n} \dfrac{1}{n + r}$Read the answer and worked solution →$\displaystyle\int_0^1 x^2(1-x)^3\, dx = ?$ (Beta function)Read the answer and worked solution →$\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{\sin^2 x}{1 + e^x}\, dx = ?$ (Glasser-type)Read the answer and worked solution →$\displaystyle\int_0^\infty x^3 e^{-x}\, dx = ?$Read the answer and worked solution →$\displaystyle\int_1^e (\ln x)^2\, dx = ?$Read the answer and worked solution →If $f(x)$ is continuous and $\displaystyle\int_0^x f(t)\, dt = x^2(1 + x)$, then $f(2) = ?$Read the answer and worked solution →$\displaystyle\int_0^1 \dfrac{x^4(1-x)^4}{1 + x^2}\, dx = ?$ (Putnam — pinpoints $22/7 - \pi$)Read the answer and worked solution →$\displaystyle\int_0^{\pi/4} \tan^4 x\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^\pi \dfrac{x}{1 + \sin x}\, dx = ?$Read the answer and worked solution →Evaluate $\displaystyle\lim_{n\to\infty} \dfrac{1^k + 2^k + \cdots + n^k}{n^{k+1}}$ for $k > -1$:Read the answer and worked solution →$\displaystyle\int_0^{\pi/2} \dfrac{dx}{1 + \tan^3 x} = ?$Read the answer and worked solution →If $\displaystyle f(x) = \int_0^x t \sin t\, dt$, then $f''(\pi) = ?$Read the answer and worked solution →Given $\displaystyle\int_0^{\pi/2} \dfrac{\sin^n x}{\sin^n x + \cos^n x}\, dx$, the value is:Read the answer and worked solution →$\displaystyle\int_0^1 \dfrac{x - 1}{\ln x}\, dx = ?$ (Frullani-type)Read the answer and worked solution →$\displaystyle\int_0^{\infty} \dfrac{\ln x}{1 + x^2}\, dx = ?$Read the answer and worked solution →Indefinite integral of xⁿ (n ≠ -1):Read the answer and worked solution →∫ (1/x) dx =Read the answer and worked solution →∫ e^x dx =Read the answer and worked solution →∫ sin x dx =Read the answer and worked solution →∫ cos x dx =Read the answer and worked solution →Linearity of integration: ∫ [af(x) + bg(x)] dx =Read the answer and worked solution →∫ tan x dx =Read the answer and worked solution →Integration by substitution: ∫ 2x × e^(x²) dx = (let u = x²)Read the answer and worked solution →∫ from 0 to π/2 sin x dx =Read the answer and worked solution →Integration by parts formula:Read the answer and worked solution →∫ x e^x dx (by parts: u = x, dv = e^x dx):Read the answer and worked solution →∫ from 0 to 1 x² dx =Read the answer and worked solution →∫ ln x dx (by parts: u = ln x, dv = dx):Read the answer and worked solution →∫ from -a to a (odd function) dx =Read the answer and worked solution →Fundamental theorem of calculus: ∫ from a to b f'(x) dx =Read the answer and worked solution →

Worked practice questions

Evaluate $\displaystyle\lim_{n\to\infty} \sum_{r=1}^{n} \dfrac{1}{n + r}$

  1. $0$
  2. $\ln 2$
  3. $\ln(1/2)$
  4. $1$
Show answer and solution

Correct answer: B. $\ln 2$

**Rewrite:** $\sum (1/n)\cdot 1/(1 + r/n) \to \int_0^1 dx/(1+x) = \ln(1+x)|_0^1 = \ln 2$. (Riemann sum recognition.)

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$\displaystyle\int_0^1 x^2(1-x)^3\, dx = ?$ (Beta function)

  1. $1/60$
  2. $1/12$
  3. $1/24$
  4. $1/720$
Show answer and solution

Correct answer: A. $1/60$

$B(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1}dx = (m-1)!(n-1)!/(m+n-1)!$. Here $m = 3$, $n = 4$: $B(3,4) = 2! \cdot 3! / 6! = 12/720 = 1/60$.

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$\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{\sin^2 x}{1 + e^x}\, dx = ?$ (Glasser-type)

  1. $\pi/4$
  2. $\pi/2$
  3. $0$
  4. $\pi$
Show answer and solution

Correct answer: A. $\pi/4$

Let $I = \int_{-\pi/2}^{\pi/2} \sin^2 x/(1 + e^x)\, dx$. Sub $x \to -x$: $I = \int_{-\pi/2}^{\pi/2} \sin^2 x/(1 + e^{-x})\, dx = \int \sin^2 x \cdot e^x/(1 + e^x)\, dx$. **Add:** $2I = \int_{-\pi/2}^{\pi/2} \sin^2 x\, dx = \pi/2$ (even integrand, area = $2 \cdot \pi/4 = \pi/2$). **So $I = \pi/4$.**

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