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AP Calculus AB Unit 6B: Definite Integration & Riemann Sums — practice questions — page 2

Explore 70 free Unit 6B: Definite Integration & Riemann Sums questions in AP Calculus AB 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 31–60 of 70.

$\displaystyle\int_{-\pi/2}^{\pi/2} \dfrac{\sin^2 x}{1 + e^x}\, dx = ?$ (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 →∫ 1/(1 + x²) dx =Read the answer and worked solution →∫ from 0 to π sin²x dx =Read the answer and worked solution →

Worked practice questions

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

  1. $\pi/4$
  2. $\pi/2$
  3. $0$
  4. $\pi$
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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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$\displaystyle\int_0^\infty x^3 e^{-x}\, dx = ?$

  1. $3!$
  2. $3$
  3. $e^{-3}$
  4. $\infty$
Show answer and solution

Correct answer: A. $3!$

**Gamma:** $\Gamma(n+1) = \int_0^\infty x^n e^{-x}\, dx = n!$. Here $n = 3$: result $= 3! = 6$.

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$\displaystyle\int_1^e (\ln x)^2\, dx = ?$

  1. $e - 2$
  2. $e + 2$
  3. $e^2 - 1$
  4. $2e - 1$
Show answer and solution

Correct answer: A. $e - 2$

**IBP:** $u = (\ln x)^2$, $dv = dx$. $= [x(\ln x)^2]_1^e - \int_1^e 2\ln x\, dx = e \cdot 1 - 0 - 2[x\ln x - x]_1^e = e - 2[(e - e) - (0 - 1)] = e - 2(1) = e - 2$.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite Limits