Definite integration practice · Easy

The value of $\int\limits_{{\pi / 4}}^{\pi / 2} \cot \theta \operatorname{cosec}^2 \theta \mathrm{d} \theta$ is

The value of the integral $\int \nolimits_{ - a}^a \,\,\frac{{x{e^x}^2}}{{1 + {x^2}}}\,\,dx$ is

The value of $\int\limits_{0}^{\pi / 2} \frac{e^{x^2}}{e^{x^2}+e^{\left(\frac{\pi}{2}-x\right)^2}} d x$ is

If $I=\int\limits_{0}^{\pi / 4} \sin ^2 x d x$ and $J=\int\limits_{0}^{\pi / 4} \cos ^2 x d x$, then $I=$

$\int \nolimits_0^{\pi /2} \,\,\frac{{dx}}{{1 + \,\,{{\tan }^3}\,\,x}}$ is equal to

$\int\limits_{{\pi / 3}}^{\pi / 2} x \sin (\pi[x]-x) d x$ is equal to :

lim \limits_ n → ∞ 1+2⁴+3⁴+… n⁴n⁵-lim _{n → ∞} 1+2³+3³+… n³n⁵$\lim \limits_\limits {n \rightarrow \infty} \frac{1+2^4+3^4+\ldots n^4}{n^5}-\lim \limits_{{n \rightarrow \infty}} \frac{1+2^3+3^3+\ldots n^3}{n^5}$

$\int \nolimits_0^{\pi /2} \,\,x\,\,\sin \,\,x\,\,dx$ is equal to

If $k=\int_0^1 \frac{e^t}{1+t} d t$, then $\int_0^1 e^t \log _e(1+t) d t$ is equal to

The value of $\int \limits_{\pi /4}^{3\pi /4} \,\,\frac{x}{{1\,\, + \,\,\sin \,\,x}}\,\,dx$ is equal to

The value of the integral $\int\limits_{{-1}}^1 \frac{x^7-3 x^5+7 x^3-x}{\cos ^2 x} d x$ is

Let a be a positive real number such that $\int_\mathrm{0}^\mathrm{a} {{\mathrm{e}^{x - \left[ x \right]}}dx} \mathrm{ = 10e - 9}\,$, where [x] is the greatest integer less than or equal to x. Then, a is equal to

Solve $\int\limits_{0}^{\pi / 2} \frac{d x}{\left(a^2 \cos ^2 x+b^2 \sin ^2 x\right)^2}$

$\int \nolimits_2^3 \,\,\frac{{dx}}{{{x^2} - x}}$ is equal to

$\lim \limits_{\mathrm{n} \rightarrow \infty} \frac{2^{\mathrm{k}}+4^{\mathrm{k}}+6^{\mathrm{k}}+\ldots+(2 \mathrm{n})^{\mathrm{k}}}{\mathrm{n}^{\mathrm{k}+1}}, \mathrm{k} \neq-1$, is equal to

If $\left| {\int_a^b f (x)dx} \right| = \int_a^b | f(x)|dx,\,\,a < b,$ then $f(x)=0$ has

The value of $\int \limits_0^1 \,\,\frac{{{2^{2x + 1}}\,\, - \,\,{5^{2x - 1}}}}{{{{10}^x}}}\,\,dx$, is

Solution of the equation $\int\limits_{{\log _e 2}}^x \frac{d t}{\sqrt{e^t-1}}=\frac{\pi}{6}$ is

lim \limits_\limitsn → ∞(nn²+1²+nn²+2²+nn²+3²+… .+15 n)$\lim \limits_\limits{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots .+\frac{1}{5 n}\right)$ is equal to:

The value of $\int\limits_{0}^{100}\left[\tan ^{-1} x\right] d x$, is

If $I\left( {m,\;\,n} \right) = \int \nolimits_0^1 \,\,{t^m}{\left( {1 + \,\,t} \right)^n}\;\,dt$, then the expression for $I\left( {m,\;\,n} \right)$ in terms of $I\left( {m + 1,\,\,\;n - 1} \right)$ is

$\lim\limits_{n \to \infty } \,\,\sum \nolimits_{r = 1}^n \,\,\frac{1}{n}\,\,{e^{r/n}}\,\,\mathrm{is}$

If $I_1=\int\limits_{x}^1 \frac{1}{1+t^2} d t$ and $I_2=\int\limits_{1}^1 \frac{1}{1+t^2} d t$ for $x>0$, then

Let [.] denote the greatest integer function, then the value of $\int\limits_{0}^{1.5} x\left[x^2\right] d x$, is

The value of $\int \nolimits_{ - 1}^{15} \,\,\mathrm{sgn}\,\,\left( {\left\{ x \right\}} \right)\,\,dx$, where $\left\{ {\, \cdot \,} \right\}$ denotes the fractional part function, is

A particle is moving on the curve with the equation $y=2 \sin ^{-1}(3 x)$. When $y=\frac{\pi}{3}$ the rate of change of " $y$ " w.r.t time is 2 unit/sec. What is the rate of change of " $x$ " coordinate of the particle w.r.t time at that instant is unit/sec

If $\int\limits_{0}^{\infty} \frac{d x}{\left(x^2+4\right)\left(x^2+9\right)}=k \pi$ then the value of $k$ is

The numbers $A, B$ and $C$ such that a function of the form $f(x)=A x^2+B x+C$ satisfies the conditions $f^{\prime}(1)=8, f(2)+f^{\prime \prime}(2)=33$ and $\int\limits_{0}^1 f(x) \mathrm{d} x=7 / 3$, are

If [.] denotes the greatest integer function, then the value of $\int\limits_{1}^3\left[x^2-2 x-2\right] d x$ is equal to

The equal to integral $\int\limits_{0}^1 \log (\sqrt{1-x}+\sqrt{1+x}) \mathrm{d} x$ is