91Ó°ÊÓ

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \sqrt{x} e^{\sqrt{x}} d x $$

Evaluate the integrals. $$ \int \frac{x^{3}}{x^{2}+x-6} d x $$

Evaluate the integral. \(\int \frac{\tan ^{3} x}{\sec ^{4} x} d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int_{1}^{e} \frac{\ln x}{x} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{1}^{\infty} \frac{1}{x(x+1)} d x\)

Evaluate the integral. \(\int \tan ^{8} x \sec ^{4} x d x\)

Evaluate the integral. \(\int \frac{\sin x}{\cos ^{2} x} d x\)

Evaluate the integral. In many cases it will be advantageous to begin by doing a substitution. For example, in Problem 19, let \(w=\sqrt{x}, w^{2}=x ;\) then \(2 w d w=d x .\) This eliminates \(\sqrt{x}\) by replacing \(x\) with a perfect square. $$ \int \frac{\ln x}{\sqrt{x}} d x $$

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{0}^{\infty} \frac{1}{x(x+1)} d x\)

Determine whether the integral is convergent or divergent. Evaluate all convergent integrals. Be efficient. If \(\lim _{x \rightarrow \infty} \neq 0\), then \(\int_{a}^{\infty} f(x) d x\) is divergent. \(\int_{e}^{\infty} \frac{1}{x \ln x} d x\)