Question

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading. (a) Three integrals that can be rewritten by using the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$so that integration by substitution applies. (b) Three integrals that can be rewritten by using the Pythagorean identity ${\tan }^{2}x+1=se{c}^{2}x$so that integration by substitution applies. (c) Three integrals that can be improved by using one or both of the double-angle identities.

Short answer

(a) $\int \frac{1}{\sqrt{a^2-x^2}}dx$, $\int \frac{1}{4-9x^2}dx$, $\int \frac{x^2}{\sqrt{16-x^2}}dx$

(b) $\int \frac{1}{1+4x^2}dx,\int \frac{1}{4+9x^2}dx,\int \frac{1}{1+x^2}dx$

(c) $\int {\sin }^{2}xdx,\int {\cos }^{2}xdx,\int {\sin }^{2}5xdx$.

Step-by-step solution

Step 1. Given Information.

The pythagorean identity is :

$$\begin{gathered} {\sin }^{2}x+{\cos }^{2}x=1 \\ {\tan }^{2}x+1=se{c}^{2}x \end{gathered}$$.

(a) Step 2. Integral Examples.

The integral is :

a) $\int \frac{1}{\sqrt{a^2-x^2}}dx$

b) $\int \frac{1}{4-9x^2}dx$

c)$\int \frac{x^2}{\sqrt{16-x^2}}dx$.

(b) Step 3. Integral example.

The examples of integral are :

$$\begin{gathered} \int \frac{1}{1+4x^2}dx \\ \int \frac{1}{4+9x^2}dx \\ \int \frac{1}{1+x^2}dx. \end{gathered}$$

(c) Step 4. Integral Example.

The examples of integral are :

$$\begin{gathered} \int {\sin }^{2}xdx \\ \int {\cos }^{2}xdx \\ \int {\sin }^{2}5xdx \end{gathered}$$.