Chapter 17

Find \(\int 3 \mathrm{e}^{2 x} \mathrm{~d} x\).

Find \(\int_{0}^{\pi / 2} \cos ^{2} t \mathrm{~d} t\).

Find the area under \(y=\cos 2 t\) from \(t=0\) to \(t=0.5\)

Use Simpson's rule with the number of strips specified to approximate the following definite integrals: (a) \(\int_{0}^{0.8} \tan ^{2} x \mathrm{~d} x, 8\) strips (b) \(\int_{1}^{2} \sqrt{1+x^{3}} \mathrm{~d} x, 10\) strips

Find \(\int_{0}^{\infty} \mathrm{e}^{-2 x} \mathrm{~d} x\)

By expressing the following in partial fractions evaluate the given integral. Remember to select the correct form for the partial fractions. $$ \int \frac{13 x-4}{6 x^{2}-x-2} \mathrm{~d} x $$

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 \(x=3\).

Find \(\int \frac{x+\cos 2 x}{3} \mathrm{~d} x\).

By writing \(\ln x\) as \(1 \times \ln x\) find \(\int \ln x \mathrm{~d} x\).

Find \(\int_{2}^{\infty} \mathrm{e}^{-3 x} \mathrm{~d} x\).