91Ó°ÊÓ

Evaluate the following integrals : (i) \(\int \mathrm{e}^{\mathrm{x}}(\sin \mathrm{x}-\cos \mathrm{x}) \mathrm{dx}\) (ii) \(\int \mathrm{e}^{\mathrm{x}}(\tan \mathrm{x}-\ln \cos x) \mathrm{dx}\) (iii) \(\int \mathrm{e}^{x} \sec x \cdot(1+\tan x) d x\)

Derive the reduction formula \(\int \cos ^{n} x d x=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x\).

\(\int x^{2} e^{3 x} d x\)

Evaluate the following integrals: (i) \(\int \frac{\mathrm{dx}}{(2 \mathrm{x}+1) \sqrt{(4 \mathrm{x}+3)}}\) (ii) \(\int \frac{1}{(x-3) \sqrt{x+1}} \mathrm{dx}\)

Find polynomials \(\mathrm{P}\) and \(\mathrm{Q}\) such that \(\int\\{(3 x-1) \cos x+(1-2 x) \sin x\\} d x=P \cos x+Q \sin x+C\)

Evaluate the following integrals: (i) \(\int \frac{\left(3 x^{2}-2\right) d x}{x^{4}-3 x^{2}-4}\) (ii) \(\int \frac{x^{2} d x}{\left(x^{2}+1\right)\left(2 x^{2}+1\right)}\) (iii) \(\int \frac{x^{2} d x}{\left(a^{2}-x^{2}\right)^{2}}\) (iv) \(\int \frac{d x}{\left(x^{2}-4 x+4\right)\left(x^{2}-4 x+5\right)}\)

Evaluate the following integrals : $$ \int \frac{x^{5} d x}{\left(1+x^{3}\right)^{1 / 2}} $$

Evaluate the following integrals : $$\int \frac{d x}{x-\sqrt{x^{2}+2 x+4}}$$

Two of these antiderivatives are elementary functions; find them. (A) \(\int \ln x d x\) (B) \(\int \frac{\ln x d x}{x}\) (C) \(\int \frac{d x}{\ln x}\)

Evaluate the following integrals: (i) \(\int \cos ^{2} x \sin ^{3} x d x\) (ii) \(\int \frac{\sin ^{3} \theta \mathrm{d} \theta}{\sqrt{\cos \theta}}\) (iii) \(\int \frac{\mathrm{d} \theta}{\sin \theta \cos ^{3} \theta}\) (iv) \(\int \frac{d x}{\sin ^{3} x \cos ^{5} x}\)