91Ó°ÊÓ

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C=0 )\) $$\int \sin 3 x \sin 6 x d x$$

\(1-80\) Evaluate the integral. $$\int x^{2} \sinh m x d x$$

\(49-54\) Use the Comparison Theorem to determine whether the integral is convergent or divergent. $$\int_{0}^{\pi} \frac{\sin ^{2} x}{\sqrt{x}} d x$$

\(53-54\) Find the area of the region bounded by the given curves. $$y=5 \ln x, \quad y=x \ln x$$

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C=0 )\) $$\int \sec ^{4} \frac{x}{2} d x$$

\(1-80\) Evaluate the integral. $$\int(x+\sin x)^{2} d x$$

\(1-80\) Evaluate the integral. $$\int \frac{d x}{x+x \sqrt{x}}$$

The integral $$\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x$$ is improper for two reasons: The interval \([0, \infty)\) is infinite and the integrand has an infinite discontinuity at \(0 .\) Evaluate it by expressing it as a sum of improper integrals of Type 2 and Type 1 as follows: $$\int_{0}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x=\int_{0}^{1} \frac{1}{\sqrt{x}(1+x)} d x+\int_{1}^{\infty} \frac{1}{\sqrt{x}(1+x)} d x$$

Find the average value of the function \(f(x)=\sin ^{2} x \cos ^{3} x\) on the interval \([-\pi, \pi] .\)

\(1-80\) Evaluate the integral. $$\int \frac{d x}{\sqrt{x}+x \sqrt{x}}$$