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AP Calculus AB Unit 6A: Indefinite Integrals & Fundamental Theorem — practice questions — page 3

Explore 84 free Unit 6A: Indefinite Integrals & Fundamental Theorem 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 6A: Indefinite Integrals & Fundamental Theorem questions and exercises with answers

Showing questions 61–84 of 84.

$\displaystyle\int_0^{\pi/2} \sin^2 x\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^\pi x \sin x\, dx = ?$Read the answer and worked solution →$\displaystyle\int_1^e \ln x\, dx = ?$Read the answer and worked solution →If $\displaystyle I_n = \int_0^{\pi/2} \sin^n x\, dx$, then $I_5 = ?$Read the answer and worked solution →Evaluate $\displaystyle\int_0^{10\pi} |\sin x|\, dx$:Read the answer and worked solution →$\dfrac{d}{dx}\left[\displaystyle\int_0^{x^2} \sin(t^2)\, dt\right] = ?$Read the answer and worked solution →$\displaystyle\int_0^{\pi/2} \dfrac{\sin x}{\sin x + \cos x}\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^{\pi/2} \log(\sin x)\, dx = ?$ (classic)Read the answer and worked solution →$\displaystyle\int_0^{\pi/2} \log(\tan x)\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^\pi \dfrac{x\sin x}{1 + \cos^2 x}\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^1 \dfrac{\ln(1+x)}{1+x^2}\, dx = ?$ (Putnam classic)Read the answer and worked solution →$\displaystyle\int x^n\,dx$ (where $n \ne -1$) 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 →$\displaystyle\int e^x\bigl(f(x) + f'(x)\bigr)\,dx$ equals:Read the answer and worked solution →Using the substitution $t = 1 + x^2$, $\displaystyle\int \dfrac{x}{1 + x^2}\,dx$ equals:Read the answer and worked solution →The number of arbitrary constants in the general solution of a differential equation of order $n$ is:Read the answer and worked solution →The integrating factor of the linear differential equation $\dfrac{dy}{dx} + \dfrac{y}{x} = x^2$ is:Read the answer and worked solution →

Worked practice questions

$\displaystyle\int_0^{\pi/2} \sin^2 x\, dx = ?$

  1. $\pi/4$
  2. $\pi/2$
  3. $1$
  4. $\pi^2/8$
Show answer and solution

Correct answer: A. $\pi/4$

$\sin^2 x = (1 - \cos 2x)/2$. $\int_0^{\pi/2} (1 - \cos 2x)/2\, dx = \tfrac{1}{2}[x - \tfrac{1}{2}\sin 2x]_0^{\pi/2} = \tfrac{1}{2}(\pi/2 - 0) = \pi/4$.

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$\displaystyle\int_0^\pi x \sin x\, dx = ?$

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

Correct answer: B. $\pi$

**Integration by parts:** $u = x$, $dv = \sin x\, dx$. $[-x\cos x]_0^\pi + \int_0^\pi \cos x\, dx = \pi + [\sin x]_0^\pi = \pi + 0 = \pi$. (Or use $\int_0^\pi x f(\sin x)\, dx = (\pi/2)\int_0^\pi f(\sin x)\, dx$ rule.)

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

  1. $0$
  2. $1$
  3. $e - 1$
  4. $e$
Show answer and solution

Correct answer: B. $1$

IBP: $[x\ln x - x]_1^e = (e - e) - (0 - 1) = 0 + 1 = 1$.

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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 1B3: Asymptotic Behavior and Infinite LimitsUnit 1B1: Limit Concepts and Graphical Limits