91Ó°ÊÓ

Evaluate the following integrals: (i) \(\int \frac{\cos \left(1-\frac{x}{2}\right)}{\sin ^{2}\left(1-\frac{x}{2}\right)} d x\) \(\int \frac{1}{(i i)} \int \frac{1}{\sqrt{\left(3-\frac{x^{2}}{4}\right)}} d x\) (iii) \(\int \frac{1}{3+(2-3 x)^{2}} d x\)

Evaluate the following integrals : (i) \(\int \frac{\cos x}{\sqrt{1+\cos x}} d x\) (ii) \(\int \frac{\mathrm{dx}}{\sin \mathrm{x} \sin (\mathrm{x}+\alpha)}\) (iii) \(\int\\{1+\cot (x-\alpha) \cot (x+\alpha\\} d x\)

Evaluate the following integrals: (i) \(\int \frac{d x}{x^{2}\left(1+x^{5}\right)^{4 / 5}}\) (ii) \(\int \frac{x^{2}-1}{x \sqrt{\left(1+x^{4}\right)}} \mathrm{dx}\)

Evaluate the following integrals: $$ \int \frac{d x}{x\left[(\log x)^{2}+2 \log x-3\right]} $$

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

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

Integrate the following functions: (i) \(\frac{1}{3} \sin \frac{2-5 x}{3}+\frac{2}{5} \cos \frac{3-2 x}{5}+\frac{4}{1+2 x}+\frac{1}{\sqrt{(3 x+1}}\) (ii) \((7-4 x)^{3}+\frac{7}{(3-7 x)}-4 \operatorname{cosec}^{2}(4 x+3)+\frac{2}{16+9 x^{2}}\)

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

Let \(f(0)=0\) and \(f^{\prime}(x)=\frac{1}{\sqrt{\left(1-x^{2}\right)}}\) for \(-1

Evaluate the following integrals: (a) Evaluate the integral \(\int \frac{1}{\sqrt{2 x-x^{2}}} d x\) in three ways: using the substitution \(u=\sqrt{x}\), using the substitution \(u=\sqrt{2-x}\), and completing the square. (b) Show that the answers in part (a) are equivalent. by comparing the coefficients.