Chapter 17

Find \(\int\left(\cos ^{2} \theta+\sin ^{2} \theta\right) \mathrm{d} \theta\).

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{1}{(x+1)(x-5)} \mathrm{d} x $$

Find the total area enclosed between the \(x\) axis and the curve \(y=x^{3}\) between \(x=-1\) and \(x=1 .\)

Find the area enclosed by \(y=4-x^{2}\) and the \(x\) axis from (a) \(x=0\) to \(x=2\), (b) \(x=-2\) to \(x=1\), (c) \(x=1\) to \(x=3\).

Use the identity \(\sin (A+B)+\sin (A-B)=2 \sin A \cos B\) to find \(\int \sin 3 x \cos 2 x \mathrm{~d} x\).

Find \(\int 7 x^{-2} \mathrm{~d} x\).

Find \(\int_{0}^{1} \frac{1+2 x}{1+x^{2}} \mathrm{~d} x\).

Find \(\int_{-\infty}^{0} \mathrm{e}^{7 x} \mathrm{~d} x\)

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{2 x}{(x-1)^{2}(x+1)} \mathrm{d} x $$

Find \(\int \frac{\mathrm{d} x}{(1-x) \sqrt{x}}\).