Chapter 2

For the following exercises, evaluate the integral using the specified method.$\int \frac{1}{x^2\sqrt{x^2+16}}dx$ using trigonometric substitution

For the following exercises, evaluate the integral using the specified method.$\int \sqrt{x}\ln (x)dx$ using integration by parts

For the following exercises, use this information: The inner product of two functions $f$ and $g$ over $[a,b]$ is defined by $f(x)\cdot g(x)=⟨f,g⟩={\int }_a^{b}f\cdot gdx$. Two distinct functions $f$ and $g$ are said to be orthogonal if $⟨f,g⟩=0$. Evaluate ${\int }_{-\pi }^{\pi }\sin (mx)\cos (nx)dx$

For the following exercises, use this information: The inner product of two functions $f$ and $g$ over $[a,b]$ is defined by $f(x)\cdot g(x)=⟨f,g⟩={\int }_a^{b}f\cdot gdx$. Two distinct functions $f$ and $g$ are said to be orthogonal if $⟨f,g⟩=0$. Integrate $y^{\prime }=\sqrt{\tan x}{\sec }^{4}x$

For the following exercises, evaluate the integral using the specified method.$\int \frac{3x}{x^3+2x^2-5x-6}dx$ using partial fractions

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. $$\int {\sin }^{456}x\cos xdx\text{ or }\int {\sin }^{2}x{\cos }^{2}xdx$$

For the following exercises, evaluate the integral using the specified method.$\int \frac{x^5}{{(4x^2+4)}^{5/2}}dx$ using trigonometric substitution

For the following exercises, evaluate the integral using the specified method.$\int \frac{\sqrt{4-{\sin }^2(x)}}{{\sin }^2(x)}\cos (x)dx$ using a table of integrals or a CAS

For each pair of integrals, determine which one is more difficult to evaluate. Explain your reasoning. $$\int {\tan }^{350}x{\sec }^{2}xdx\text{ or }\int {\tan }^{350}x\sec xdx$$

For the following exercises, integrate using whatever method you choose.$\int {\sin }^2(x){\cos }^2(x)dx$