91Ó°ÊÓ

$$\text { Prove that } \int \csc x d x=-\ln |\csc x+\cot x|+C$$

Use a computer algebra system to evaluate the following indefinite integrals. Assume that a is a positive real number. $$\int \frac{x}{\sqrt{2 x+3}} d x$$

Evaluate the following definite integrals. $$\int_{0}^{1} \frac{d x}{\sqrt{x^{2}+16}}$$

Evaluate the following integrals or state that they diverge. $$\int_{0}^{1} \ln x^{2} d x$$

Use the approaches discussed in this section to evaluate the following integrals. $$\int \frac{d x}{x^{1 / 2}+x^{3 / 2}}$$

A standard pendulum of length \(L\) swinging under only the influence of gravity (no resistance) has a period of $$T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin ^{2} \varphi}}$$ where \(\omega^{2}=g / L, k^{2}=\sin ^{2}\left(\theta_{0} / 2\right), g \approx 9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(\theta_{0}\) is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with \(L=1 \mathrm{m}\) that is released from an angle of \(\theta_{0}=\pi / 4\) rad.

Evaluate the following integrals. $$\int \frac{2 x+1}{x^{2}+4} d x$$

Evaluate the following definite integrals. $$\int_{8 \sqrt{2}}^{16} \frac{d x}{\sqrt{x^{2}-64}}$$

Use the approaches discussed in this section to evaluate the following integrals. $$\int_{0}^{1} \frac{d x}{4-\sqrt{x}}$$

Evaluate the following integrals or state that they diverge. $$\int_{-1}^{1} \frac{x}{x^{2}+2 x+1} d x$$