Question

integrate in problems 12 - 16.

1. (int_{2}^{12} \frac{2 x}{3} sqrt{x^{2}+1} d x)
2. (int(cos x) e^{sin x} d x)

Explanation:

Step1: Use substitution method

Let \(u = x^{2}+1\), then \(du=2xdx\).
The integral \(\int\frac{2x}{3}\sqrt{x^{2}+1}dx=\frac{1}{3}\int\sqrt{u}du\).

Step2: Integrate \(\sqrt{u}\)

We know that \(\int u^{n}du=\frac{u^{n + 1}}{n+1}+C\) (\(n
eq - 1\)). For \(n=\frac{1}{2}\), \(\int\sqrt{u}du=\int u^{\frac{1}{2}}du=\frac{2}{3}u^{\frac{3}{2}}+C\).
So \(\frac{1}{3}\int\sqrt{u}du=\frac{1}{3}\times\frac{2}{3}u^{\frac{3}{2}}+C=\frac{2}{9}(x^{2}+1)^{\frac{3}{2}}+C\).

Answer:

\(\frac{2}{9}(x^{2}+1)^{\frac{3}{2}}+C\)