Integral Calculus Questions (265)

If $f(x)=\displaystyle\int x^{1/3}(x^{2/3}+x^{1/3}+1)(2x^{2/3}+3x^{1/3}+6)^3\,dx$, $f(0)=0$, then $\dfrac{16}{11}f(1)$ is equal to

For positive integer $n$, let $I_n=\displaystyle\int_{-\pi}^{\pi}\!\left(\frac{\pi}{2}-|x|\right)\cos nx\,dx$. Find $[I_1+I_2+I_3+I_4]$ (GIF).

Let $f$ be a function on $(0,2\pi]$ such that $\displaystyle\int_0^x (f'(t)-\sin 2t)\,dt = \int_x^0 f(t)\tan t\,dt$ and $f(0)=1$. If the maximum value of $f(x)$ is $m$, then $8m$ equals

If $f(x)=\displaystyle\int x^{1/3}(x^{2/3}+x^{1/3}+1)(2x^{2/3}+3x^{1/3}+6)^3\,dx$, $f(0)=0$, then $\dfrac{16}{11}f(1)$ is equal to

Let $I=\displaystyle\int\frac{\sin^2\theta\cdot\log_e(e\cot\theta)}{\cos^4\theta((\log_e\tan\theta)^4-(\tan\theta)^4)}d\theta=\frac{1}{4}\log_e|f(\theta)|-\frac{1}{2}\tan^{-1}(g(\theta))+c$. The value of $f(\theta)-g(\theta)$ at $\theta=\pi/3$ is

If $\displaystyle\int\frac{\sec^2 x-2010}{\sin^{2010}x}\,dx=\frac{P(x)}{(\sin x)^{2010}}+C$, then the value of $P\!\left(\dfrac{\pi}{3}\right)$ is

$\displaystyle\int_0^{\pi/2}\dfrac{dx}{a^2\cos^2x+b^2\sin^2x}=$

If $f(x) = \begin{vmatrix}\cos x & e^{x^2} & 2x\cos^2(x/2)\\ x^2 & \sec x & \sin x+x^3\\ 1 & 2 & x+\tan x\end{vmatrix}$ and $\displaystyle\int_{-\pi/2}^{\pi/2}(1+x^4)(f(x)+f''(x))\,dx = 2\lambda+3$, then $\lambda$ is

$f(x)=\min(x-[x],1-x+[x])$ for $x\in\mathbb{R}$. Then $\displaystyle 2\int_0^2 f(x)dx=$

$\displaystyle\int_{\pi/4}^{3\pi/4}\dfrac{x}{1+\sin x}dx$

The area enclosed by the curves $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $\left[0,\dfrac{\pi}{2}\right]$ is

If $\displaystyle\int (x^2+1)(x+1)^2 e^x\,dx = A(f(x))^2 + C$ and $f(-1) = \frac{2}{e}$, then $2A + f(0)$ equals

The area bounded by the $x$-axis, part of the curve $y=1+x^{-2}$ and the ordinates $x=1$, $x=2$ is divided into equal parts by the ordinate at $x=a$ such that $a=\dfrac{3+\sqrt{p}}{8}$. The value of $p$ is

Let $f:(0,\infty)\to\mathbb{R}$ be a differentiable function such that $f'(x)=2-\dfrac{f(x)}{x}$ for all $x\in(0,\infty)$ and $f(1)\ne 1$. Then

Let $f$ be a function defined by $f(x)=\begin{cases}(x-r)^2, & r-1\leq x<r+1\\ 1, & r+1\leq x\leq r+2\end{cases}$ where $r=3k$, $k\in I$. Find $\displaystyle\sqrt{\int_0^{45}f(x)\,dx}$