Reduction Formulae

Find the integral of $\int x^2\cos 2xdx$ using reduction formula.

Find the integral of $\int x^2\cos xdx$ using reduction formula.

Find the integral of $\int x^2\sin 4xdx$ using reduction formula.

Find the integral of $\int x^2\sin 2xdx$ using reduction formula.

Find the integral of $\int x^2\sin xdx$ using reduction formula.

Find the reduction formula for $\int \frac{\cos x}{x^n}dx$.

Find the reduction formula for $\int \frac{\cos ax}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin 3x}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin x}{x^n}dx$.

Find the reduction formula for $\int \frac{\sin ax}{x^n}dx$.

If $I_n=\int {\left(\log x\right)}^{n}dx$, then $I_n+n I_{n-1}$ $=$?

If $l_n=\int {\tan }^{n}xdx$ and $l_n=-\frac{{\tan }^{n-1}x}{(n-1)}+\lambda l_{n-2}$, then $\lambda$ is equal to:

If $I_n= \int {\left(lnx\right)}^n dx,$ then $I_{10}+10.I_9=$

For $x>0$, if $\int {\left(\log x\right)}^{5}dx=x\left[A{\left(\log x\right)}^5+B{\left(\log x\right)}^4+C{\left(\log x\right)}^3+D{\left(\log x\right)}^2+E\left(\log x\right)+F\right]+$constant, then the value of $A+B+C+D+E+F$ is equal to

For any integer $n \ge 2$, let $I_n=\int {\tan }^{n}xdx$. If $I_n=\frac{1}{a}{\tan }^{n-1}x-b{I}_{n-2}+c$, then the ordered pair $\left(a,b\right)$, equals to

Let ${\mathrm{I}}_{\mathrm{m},\mathrm{n}}=\int {\cos }^{\mathrm{m}}\mathrm{x}\sin \mathrm{n}\mathrm{x}\mathrm{ }\mathrm{d}\mathrm{x}$ then $7{\mathrm{I}}_{4,\mathrm{ }3}-4{\mathrm{I}}_{3,\mathrm{ }2}$  is equal to

If ${\mathrm{I}}_{\mathrm{n}}=\int {\left(\log \mathrm{x}\right)}^{\mathrm{n}}\mathrm{d}\mathrm{x}$ then ${\mathrm{I}}_{\mathrm{n}}+\mathrm{n}\mathrm{ }{\mathrm{I}}_{\mathrm{n}–1}$ is equal to

If ${\mathrm{I}}_{\mathrm{m},\mathrm{ }\mathrm{n}}=\int {\cos }^{\mathrm{m}}\mathrm{x}\mathrm{ }\sin \mathrm{n}\mathrm{x}\mathrm{ }\mathrm{d}\mathrm{x}$ , then $7{\mathrm{I}}_{4,\mathrm{ }3}-\mathrm{ }4{\mathrm{I}}_{3,\mathrm{ }2}=$

If ${\mathrm{I}}_{\mathrm{n}}=\int {\cot }^{\mathrm{n}}\mathrm{x}\mathrm{ }\mathrm{d}\mathrm{x}$ , then ${\mathrm{I}}_0+{\mathrm{I}}_1+2\left({\mathrm{I}}_2+{\mathrm{I}}_3\right.$

$+\mathrm{ }\dots .+\left.{\mathrm{I}}_8\right)+{\mathrm{I}}_9\mathrm{ }+{\mathrm{I}}_{10}=$

If ${\mathrm{u}}_{\mathrm{n}}=\int x^{\mathrm{n}}\sqrt{\left({\mathrm{a}}^2-{\mathrm{x}}^2\right)}\mathrm{d}\mathrm{x}$ , prove that

${\mathrm{u}}_{\mathrm{n}}=-\mathrm{ }\frac{{\mathrm{x}}^{\mathrm{n}-1}{\left({\mathrm{a}}^2-{\mathrm{x}}^2\right)}^{3/2}}{\left(\mathrm{n}+2\right)}+\mathrm{ }\frac{\left(\mathrm{n}-1\right)}{\left(\mathrm{n}+2\right)}\mathrm{ }{\mathrm{a}}^2\mathrm{ }{\mathrm{u}}_{\mathrm{n}–\mathrm{ }2}$