91Ó°ÊÓ

Show that \(\int \cot ^{9} x d x=-\frac{\cot ^{8} x}{8}+\frac{\cot ^{6} x}{6}-\frac{\cot ^{4} x}{4}+\frac{\cot ^{2} x}{2}\) \(+\ln |\sin x|+C\)

Applying Ostrogradsky's method, find the following integrals: (i) \(\int \frac{d x}{(x+1)^{2}\left(x^{2}+1\right)^{2}}\) (ii) \(\int \frac{d x}{\left(x^{4}+1\right)^{2}}\) (iii) \(\int \frac{\mathrm{dx}}{\left(\mathrm{x}^{2}+1\right)^{4}}\) (iv) \(\int \frac{x^{4}-2 x^{2}+2}{\left(x^{2}-2 x+2\right)^{2}} d x\)

Evaluate the following integrals: $$ \int \frac{x^{2}}{\sqrt{\left(a^{6}-x^{6}\right)}} d x $$

Evaluate the following integrals: (i) \(\int \frac{2 x^{3}+x^{2}+4}{\left(x^{2}+4\right)^{2}} d x\) (ii) \(\int \frac{x^{3}+x^{2}-5 x+15}{\left(x^{2}+5\right)\left(x^{2}+2 x+3\right)} d x\)(iii) \(\int \frac{d x}{\left(x^{4}+2 x+10\right)^{3}}\) (iv) \(\int \frac{x^{5}-x^{4}+4 x^{3}-4 x^{2}+8 x-4}{\left(x^{2}+2\right)^{3}} d x\)

Evaluate the following integrals: (i) \(\int x^{3} e^{x} d x\) (ii) \(\int x^{3} \cos x d x\) (iii) \(\int x^{3} / n^{2} x d x\)

Given the continuous periodic function \(\mathrm{f}(\mathrm{x})\), \(\mathrm{x} \in \mathrm{R}\). Can we assert that the antiderivative of that function is a periodic function ?