Definite Integration

If $I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n>0$, then $I(9,14)+I(10,13)$ is

The value of $\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}}{e^{\left(\left(\log _e x\right)^2+1\right)^{-1}}+e^{\left(\left(6-\log _e x\right)^2+1\right)^{-1}}}\right) d x$ is

If $\mathrm{I}=\int_0^{\frac{\pi}{2}} \frac{\sin ^{\frac{3}{2}} x}{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x} \mathrm{~d} x$, then $\int_0^{2I} \frac{x \sin x \cos x}{\sin ^4 x+\cos ^4 x} \mathrm{~d} x$ equals :

Let for $f(x)=7 \tan ^8 x+7 \tan ^6 x-3 \tan ^4 x-3 \tan ^2 x, \quad \mathrm{I}_1=\int_0^{\pi / 4} f(x) \mathrm{d} x$ and $\mathrm{I}_2=\int_0^{\pi / 4} x f(x) \mathrm{d} x$. Then $7 \mathrm{I}_1+12 \mathrm{I}_2$ is equal to :

Let \(\lim _\limits{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right.\) $$\left.+\ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2...

\(\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)\) is equal to

The integral \(\int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x\) is equal to

The value of the integral \(\int_\limits{-1}^2 \log _e\left(x+\sqrt{x^2+1}\right) d x\) is

The value of \(k \in \mathbb{N}\) for which the integral \(I_n=\int_0^1\left(1-x^k\right)^n d x, n \in \mathbb{N}\), satisfies \(147 I_{20}=148 I_{21}\) is

Let \(\int_\limits\alpha^{\log _e 4} \frac{\mathrm{d} x}{\sqrt{\mathrm{e}^x-1}}=\frac{\pi}{6}\). Then \(\mathrm{e}^\alpha\) and \(\mathrm{e}^{-\alpha}\) are the roots of the equation :

Let \(r_k=\frac{\int_0^1\left(1-x^7\right)^k d x}{\int_0^1\left(1-x^7\right)^{k+1} d x}, k \in \mathbb{N}\). Then the value of \(\sum_\limits{k=1}^{10} \frac{1}{7\left(r_k-1\right)}\) is equal to _________.

\(\int_\limits0^{\pi / 4} \frac{\cos ^2 x \sin ^2 x}{\left(\cos ^3 x+\sin ^3 x\right)^2} d x \text { is equal to }\)

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}\), w...

The integral \(\int_\limits0^{\pi / 4} \frac{136 \sin x}{3 \sin x+5 \cos x} \mathrm{~d} x\) is equal to :

The value of \(\int_\limits{-\pi}^\pi \frac{2 y(1+\sin y)}{1+\cos ^2 y} d y\) is :

If \(f(t)=\int_\limits0^\pi \frac{2 x \mathrm{~d} x}{1-\cos ^2 \mathrm{t} \sin ^2 x}, 0<\mathrm{t}<\pi\), then the value of \(\int_\limits0^{\frac{\pi}{2}} \frac{\pi^2 \mathrm{dt}}{f(\mathrm{t})}\) equals __________.

Let \(\beta(\mathrm{m}, \mathrm{n})=\int_\limits0^1 x^{\mathrm{m}-1}(1-x)^{\mathrm{n}-1} \mathrm{~d} x, \mathrm{~m}, \mathrm{n}>0\). If $$\int_\limits0^1\left(1-x^{10}\right)^{20} \mathrm{~d} x=\mathrm{a} \times \beta(\mathrm{b}, \mathrm{c}...

If \(\int_0^{\frac{\pi}{4}} \frac{\sin ^2 x}{1+\sin x \cos x} \mathrm{~d} x=\frac{1}{\mathrm{a}} \log _{\mathrm{e}}\left(\frac{\mathrm{a}}{3}\right)+\frac{\pi}{\mathrm{b} \sqrt{3}}\), where \(\mathrm{a}, \mathrm{b} \in \mathrm{N}\), then $$...

If the shortest distance between the lines \(\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}\) and \(\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}\) is \(\frac{38}{3 \sqrt{5}} \mathrm{k}\), and $$\int_\limits 0^{\mathrm{k}}\left[x^2\right] \mathrm{d...

$$\text { Let } f(x)=\left\{\begin{array}{lr} -2, & -2 \leq x \leq 0 \\ x-2, & 0< x \leq 2 \end{array} \text { and } \mathrm{h}(x)=f(|x|)+|f(x)| \text {. Then } \int_\limits{-2}^2 \mathrm{~h}(x) \mathrm{d} x\right. \text { is equal to: }$$

If the value of the integral \(\int\limits_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x\) is \(\frac{2}{\pi}\).Then, a value of \(\alpha\) is

Let \(f(x)=\int_0^x\left(t+\sin \left(1-e^t\right)\right) d t, x \in \mathbb{R}\). Then, \(\lim _\limits{x \rightarrow 0} \frac{f(x)}{x^3}\) is equal to

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\frac{4^x}{4^x+2}\) and \(M=\int_\limits{f(a)}^{f(1-a)} x \sin ^4(x(1-x)) d x, N=\int_\limits{f(a)}^{f(1-a)} \sin ^4(x(1-x)) d x ; a \neq \frac{1}{2}\). If $$\alph...

Let \(S=(-1, \infty)\) and \(f: S \rightarrow \mathbb{R}\) be defined as
\(f(x)=\int_\limits{-1}^x\left(e^t-1\right)^{11}(2 t-1)^5(t-2)^7(t-3)^{12}(2 t-10)^{61} d t \text {, }\)
Let \(\mathrm{p}=\) Sum of squares of the values of \(x\), whe...

If the integral \(525 \int_\limits0^{\frac{\pi}{2}} \sin 2 x \cos ^{\frac{11}{2}} x\left(1+\operatorname{Cos}^{\frac{5}{2}} x\right)^{\frac{1}{2}} d x\) is equal to \((n \sqrt{2}-64)\), then \(n\) is equal to _________.

\(\left|\frac{120}{\pi^3} \int_\limits0^\pi \frac{x^2 \sin x \cos x}{\sin ^4 x+\cos ^4 x} d x\right| \text { is equal to }\) ________.

Let \(f, g:(0, \infty) \rightarrow \mathbb{R}\) be two functions defined by \(f(x)=\int\limits_{-x}^x\left(|t|-t^2\right) e^{-t^2} d t\) and \(g(x)=\int\limits_0^{x^2} t^{1 / 2} e^{-t} d t\). Then, the value of $$9\left(f\left(\sqrt{\log _e...

The value of \(9 \int_\limits0^9\left[\sqrt{\frac{10 x}{x+1}}\right] \mathrm{d} x\), where \([t]\) denotes the greatest integer less than or equal to \(t\), is

Let \(f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R}\) be a differentiable function such that \(f(0)=\frac{1}{2}\). If the $$\lim _\limits{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t}) \mathrm{dt}}{\mathrm{e}^{x^2}-1}...

The value of \(\lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}\) is :

Let \(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) be defined as \(f(x)=a e^{2 x}+b e^x+c x\). If \(f(0)=-1, f^{\prime}\left(\log _e 2\right)=21\) and \(\int_0^{\log _e 4}(f(x)-c x) d x=\frac{39}{2}\), then the value of \(|a+b+c|\) equal...

Let \(a\) and \(b\) be real constants such that the function \(f\) defined by $$f(x)=\left\{\begin{array}{ll}x^2+3 x+a & , x \leq 1 \\ b x+2 & , x>1\end{array}\right.$$ be differentiable on \(\mathbb{R}\). Then, the value of $$\int_\limits{...

Let \(y=f(x)\) be a thrice differentiable function in \((-5,5)\). Let the tangents to the curve \(y=f(x)\) at \((1, f(1))\) and \((3, f(3))\) make angles \(\pi / 6\) and \(\pi / 4\), respectively with positive \(x\)-axis. If $$27 \int_\limi...

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\frac{x}{\left(1+x^4\right)^{1 / 4}}\), and \(g(x)=f(f(f(f(x))))\). Then, \(18 \int_0^{\sqrt{2 \sqrt{5}}} x^2 g(x) d x\) is equal to

If the value of the integral \(\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2\), then the value of \(a\) is

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)\) is equal to

If \(\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}\), where \(\alpha, \beta\) and \(\gamma\) are rational numbers, then
\(3 \alpha+4 \beta-\gamma\) is equal to _________.

Let the slope of the line \(45 x+5 y+3=0\) be \(27 r_1+\frac{9 r_2}{2}\) for some \(r_1, r_2 \in \mathbb{R}\). Then \(\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)\) is equal to ...

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $\mathrm{I}_1=\int\limits_{\mathrm{a}}^{\mathrm{b}} x \sin \left(4 x-x^2\right) \mathrm{d} x, \mathrm{I}_2=\int\limits_{\mathrm{a}}^{\mathrm{b}...

If $\int\limits_0^1 \frac{1}{\sqrt{3+x}+\sqrt{1+x}} \mathrm{~d} x=\mathrm{a}+\mathrm{b} \sqrt{2}+\mathrm{c} \sqrt{3}$, where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are rational numbers, then $2 \mathrm{a}+3 \mathrm{~b}-4 \mathrm{c}$ is equal ...

Let \(f(x)=\int_\limits0^x g(t) \log _{\mathrm{e}}\left(\frac{1-\mathrm{t}}{1+\mathrm{t}}\right) \mathrm{dt}\), where \(g\) is a continuous odd function.
If $$\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^2 \cos x}{1+\mathrm{e}^x}\right) \mat...

For \(0 < \mathrm{a} < 1\), the value of the integral \(\int_\limits0^\pi \frac{\mathrm{d} x}{1-2 \mathrm{a} \cos x+\mathrm{a}^2}\) is :

If $\int\limits_{-\pi / 2}^{\pi / 2} \frac{8 \sqrt{2} \cos x \mathrm{~d} x}{\left(1+\mathrm{e}^{\sin x}\right)\left(1+\sin ^4 x\right)}=\alpha \pi+\beta \log _{\mathrm{e}}(3+2 \sqrt{2})$, where $\alpha, \beta$ are integers, then $\alpha^2+\...

The value of the integral $\int\limits_0^{\pi / 4} \frac{x \mathrm{~d} x}{\sin ^4(2 x)+\cos ^4(2 x)}$ equals :

Let $f:(0, \infty) \rightarrow \mathbf{R}$ and $\mathrm{F}(x)=\int\limits_0^x \mathrm{t} f(\mathrm{t}) \mathrm{dt}$. If $\mathrm{F}\left(x^2\right)=x^4+x^5$, then $\sum\limits_{\mathrm{r}=1}^{12} f\left(\mathrm{r}^2\right)$ is equal to ____...

The value of $\int\limits_0^1\left(2 x^3-3 x^2-x+1\right)^{\frac{1}{3}} \mathrm{~d} x$ is equal to :

If $\int\limits_0^{\frac{\pi}{3}} \cos ^4 x \mathrm{~d} x=\mathrm{a} \pi+\mathrm{b} \sqrt{3}$, where $\mathrm{a}$ and $\mathrm{b}$ are rational numbers, then $9 \mathrm{a}+8 \mathrm{b}$ is equal to :

Let \([t]\) denote the greatest integer \(\leq t\). Then \(\frac{2}{\pi} \int_\limits{\pi / 6}^{5 \pi / 6}(8[\operatorname{cosec} x]-5[\cot x]) d x\) is equal to __________.

Let \([t]\) denote the greatest integer function. If \(\int_\limits{0}^{2.4}\left[x^{2}\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}\), then \(\alpha+\beta+\gamma+\delta\) is equal to __________.

Let \(5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0\). Then \(18 \int_\limits{1}^{2} f(x) d x\) is equal to :