allery$\displaystyle\int_0^\pi \sin x\, dx = ?$
- $0$
- $1$
- $2$
- $\pi$
Show answer and solution
Correct answer: C. $2$
$[-\cos x]_0^\pi = -\cos\pi + \cos 0 = 1 + 1 = 2$.
AP Intermediate 2nd Year Integrals — practice questions
Explore 15 free Integrals questions in AP Intermediate 2nd Year Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
Practise Integrals with Quill →Integrals questions and exercises with answers
Showing questions 1–15 of 15.
$\displaystyle\int_0^\pi \sin x\, dx = ?$Read the answer and worked solution →If $f$ is **odd** on $[-a, a]$, then $\displaystyle\int_{-a}^{a} f(x)\, dx$ equals:Read the answer and worked solution →$\displaystyle\int_0^1 \dfrac{1}{1 + x^2}\, dx = ?$Read the answer and worked solution →$\displaystyle\int_0^{\pi/2} \sin^2 x\, dx = ?$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 →$\displaystyle\int_0^2 [x]\, dx$ where $[x]$ is the greatest integer function: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 →
Worked practice questions
If $f$ is **odd** on $[-a, a]$, then $\displaystyle\int_{-a}^{a} f(x)\, dx$ equals:
- $0$
- $2\int_0^a f(x)dx$
- $a$
- Cannot be determined
Show answer and solution
Correct answer: A. $0$
By symmetry, odd-function integrals over symmetric intervals vanish.
(For even $f$: $2\int_0^a f$.)
$\displaystyle\int_0^1 \dfrac{1}{1 + x^2}\, dx = ?$
- $\pi/2$
- $\pi/4$
- $\pi$
- $1$
Show answer and solution
Correct answer: B. $\pi/4$
$[\arctan x]_0^1 = \arctan 1 - \arctan 0 = \pi/4$.
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