Recent questions tagged integration

Find the length of the curve $x=\cos t,\ y=t+\sin t,\ 0\le t\le \pi$.

Find $\dfrac{dy}{dx}$ where $$y=\int_{x^2}^{\sin x}\sqrt{1+t^4}dt$$$\cos x\sqrt{1+\sin^4x}-2x\sqrt{1+x^8}$ $\cos x\sqrt{1+\sin^4x}+2x\sqrt{1+x^8}$ $\sqrt{1+\sin^4x}-\sqrt...

Find $f(x)$ if $f'(x)=6x^2+\sec^2 x$ and $f(0)=4$.$f(x)=2x^3+\tan x+4$ $f(x)=2x^3-\tan x+4$ $f(x)=6x^3+\tan x+4$ $f(x)=2x^3+\tan x-4$

Find the area of the region bounded by the lines $x=-1$, $x=2$, $y=0$, and the curve $y=x^3-x^2-2x$.$\frac{5}{12}$ $\frac{8}{3}$ $\frac{37}{12}$ $\frac{11}{4}$

The volume of the solid generated by revolving the region bounded by the curves $y=x^2$ and $y=x+2$ about the $x$-axis is:$\frac{72}{5}$ $\frac{72\pi}{5}$ $\frac{36\pi}{5...

The area of the region between the curves $y=x^2-2x$ and $y=2x-x^2$ for $x\in[-1,3]$ is:$4$ $\frac{16}{3}$ $\frac{20}{3}$ $8$

Evaluate $$I=\int_0^1 \frac{x^2}{(1-x^2)^{3/2}}dx$$$\frac{\pi}{4}$ $1$ $2$ The integral diverges

Let $f(x)=\left|x-\lfloor x\rfloor-\frac{1}{2}\right|$, where $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$. The value of $\int_{0.2}^{2.75} f...

Let $f:[0,1]\to\mathbb{R}$ be continuous and satisfy $f(x)=x^2+\int_0^1 (x+t)f(t)dt$ for every $x\in[0,1]$. Which of the following is $f(x)$?$x^2-5x-\frac{17}{6}$ $x^2+5x...

Let $f$ be a continuous function defined for all real numbers. Suppose :$$\int_0^3 f(x)dx=4, \qquad\int_0^6 f(x)dx=9, \qquad\int_6^{10} f(x)dx=-2, \qquad \text{and} \qqua...

Evaluate the definite integral:$$I=\int_0^\pi x\sin x\ln(1+\cos^2 x),dx$$Which of the following is the correct value?$\frac{\pi^2}{2}+\pi\ln 2-2\pi$ $\frac{\pi^2}{4}+\pi\...

The value of the given integral is : $$I=\int_0^{\pi/2}\left(\frac{4\sqrt{\sin x}+7\sqrt{\cos x}}{\sqrt{\sin x}+\sqrt{\cos x}}+\sin x\cos x\right)dx$$$\frac{11\pi}{4}+\fr...

The integral $\int_0^1 \frac{d x}{\sqrt{(1-x)}}$ is equal to

The value of the following improper integral is $\int_0^1 x \ln x$$1 / 4$ $0$ $-1 / 4$ $1$

Evaluate the integral:$$\int_0^{\frac{\pi}{2}} \frac{\tan ^7 x}{\cot ^7 x+\tan ^7 x} d x$$$\frac{\pi}{4}$ $\frac{\pi}{2}$ $\frac{1}{2}$ $\frac{\pi}{8}$

If $g^{\prime}(x)=f(x)$ then $\int x^{3} f\left(x^{2}\right) d x$ is given by$x^{2} g\left(x^{2}\right)-\int x g\left(x^{2}\right) d x+C$$\frac{1}{2} x^{2} g\left(x^{2}\r...