List of top Mathematics Questions on Methods of Integration asked in MHT CET

\(\int \frac{5x^{12}+6x^{11}}{(x^{6}+x+1)^{3}}dx=\)

If \(\int (\sqrt{\tan x}+\sqrt{\cot x})dx=a\tan ^{-1}\left(\frac{\tan x-1}{\sqrt{b\tan x}}\right)+c\), for \(x\in \left(0,\frac{\pi }{2}\right)\), then \(a^{10}+b^{5}=\cdots\)

If \(\int x(a-x)^{9}dx=\frac{[f(x)]^{11}}{11}-a\frac{[f(x)]^{10}}{10}+c\), where \(c\) is a constant of integration, then \(f'(x)=\cdots\)

The integral \(\int \frac{\csc x}{\cos ^{2}[1+\log (\tan \frac{x}{2})]}\ dx\) is equal to:

\(\int \frac{x}{x^{5/2}+4x^{3/2}}\ dx=\ldots\)

If \(\int \sqrt{1+\sec x}dx=2\sin ^{-1}(f(x))+c\), then

\(\int \frac{x^{3}\sin (\tan ^{-1}(x^{4}))}{1+x^{8}}dx=\)

If \(\int \sqrt{e^{x}-1}dx=a\sqrt{e^{x}-1}+b\tan ^{-1}(\sqrt{e^{x}-1})+c\), then

\(\int f(x)dx=g(x)+c\), then \(\int xf(x^{2})dx=\)

\(\int \sqrt{x}\sec \left(x^{\frac{3}{2}}\right)\tan \left(x^{\frac{3}{2}}\right)dx=\)

If \(f(x)=\frac{1}{\sqrt{x}},g(x)=3-2x-x^{2}\) then \(\int [(f\circ g)(e^{x})]\cdot e^{x}dx=\cdots \cdots \cdots\)

If \(\int \frac{\log (\log x)}{4x(\log x)^{2}}dx=k\left[\frac{1+\log (\log x)}{\log x}\right]+c\), then the value of \(k\) is...

The value of the integral \(\int \frac{dx}{x\sqrt{1-x^{3}}}\) is equal to...

The value of the integral \(\int \frac{dx}{3+\sin 2x+\sin ^{2}x}=\) .........

The value of the integral \(\int \frac{x^{2}+1}{(x+1)^{2}}dx=\) ............

The value of the integral \(\int \frac{1}{\cos x\cdot \log \left(\tan \left(\frac{\pi }{4}+\frac{x}{2}\right)\right)}dx\) is....

If \(f(x)=\int \frac{x^{2}}{(1-x^{2})(1+\sqrt{1-x^{2}})}dx\) and \(f(0)=2\), then \(f\left(\frac{1}{2}\right)=\) ...

If \(\int \sin ^{4}x\cdot e^{\log (\cos x)}dx=\frac{1}{k}\sin ^{k}x+C\), for \(x\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)\), then ...

For \(x\neq 0\), the value of \(I=\int \frac{x^{2}-1}{(x^{4}+3x^{2}+1)\tan ^{-1}\left(x+\frac{1}{x}\right)}dx\) is ...

If \(I_{1}=\int \frac{e^{x}}{e^{2x}+e^{x}+1}dx\) and \(I_{2}=\int \frac{1}{e^{-2x}+e^{-x}+1}dx\), then \(I_{1}-I_{2}=\)

If \(f(x)\) is an antiderivative of \(\frac{x+1}{\sqrt{x-1}}\) with respect to \(x\), then value of \(f(5)-f(2)\) is...

\(\int (x^{21}+x^6+x^3)(2x^{18}+7x^3+14)^{\frac{1}{3}} dx =\)

If \(\int f(x)dx = g(x)+c\) then \(\int f^{-1}(x)dx =\)

The value of \(\int sin4xcos3x\,dx\) is

\(\int \frac{x^4(x^{10}-1)}{x^{20}+3x^{10}+1}\,dx = \cdots\)