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Chapter 1: Problem 1
\(\int x^{2} e^{3 x} d x\)
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Evaluate the following integrals : (i) \(\int \mathrm{e}^{\mathrm{x}}\left(\frac{1-\mathrm{x}}{1+\mathrm{x}^{2}}\right)^{2} \mathrm{dx}\) (ii) \(\int \mathrm{e}^{x} \frac{\left(x^{3}-x+2\right)}{\left(x^{2}+1\right)^{2}} d x\) (iii) \(\int \frac{\mathrm{e}^{\mathrm{x}}(\mathrm{x}-1)}{(\mathrm{x}+1)^{3}} \mathrm{dx}\) (iv) \(\int \mathrm{e}^{x}\left(\frac{1-x}{1+x}\right)^{2} d x\)
Evaluate the following integrals: $$ \int \frac{x^{3} d x}{\left(x^{2}-2 x+2\right)} $$
Evaluate the following integrals: (i) \(\int \mathrm{e}^{\mathrm{x}}[\ln (\sec x+\tan \mathrm{x})+\sec \mathrm{x}] \mathrm{d} \mathrm{x}\) (ii) \(\int \mathrm{e}^{x}\left(\log x+\frac{1}{x^{2}}\right) d x\)
Three of these six antiderivatives are elementary. Find them. (A) \(\int x \cos x d x\) (B) \(\int \frac{\cos x}{x} d x\) (C) \(\int \frac{x d x}{\ln x}\) (D) \(\int \frac{\ln x^{2}}{x} d x\) (E) \(\int \sqrt{x-1} \sqrt{x} \sqrt{x+1} d x\) (F) \(\int \sqrt{x-1} \sqrt{x+1} x d x\)
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) \(\int \frac{x^{4}-2 x^{2}+2}{\left(x^{2}-2 x+2\right)^{2}} d x\)
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