alleryStart learning →$\displaystyle\int_0^\pi \sin x\, dx = ?$
- $0$
- $1$
- $2$
- $\pi$
Show answer and solution
Correct answer: C. $2$
$[-\cos x]_0^\pi = -\cos\pi + \cos 0 = 1 + 1 = 2$.
CG PET Definite Integrals — practice questions
Explore 31 free Definite Integrals questions in CG PET Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
Practise Definite Integrals with Quill →Definite Integrals questions and exercises with answers
Showing questions 1–30 of 31.
$\displaystyle\int_0^\pi \sin x\, dx = ?$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 →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 →
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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