Question

Construct examples of the thing(s) described in the following.

(a) Five integrals that can be solved with the method of

integration by substitution.

(b) Five integrals that cannot be solved with the method

of integration by substitution.

(c) Three relatively simple integrals that we cannot solve

with any of the methods we now know.

Short answer

(a)

\(\begin{align*}1.&\int x^7(x^8-5)^4dx\\2.&\int\frac{e^xdx}{e^x+7}\\3.&\int \frac{(\log x)^5dx}{x}\\4.&\int\frac{\tan^{-1}xdx}{1+x^2}\\5.&\int \frac{dx}{x(x^3-1)} \end{align*}\)

(b)

\(\begin{align*}1.&\int \sqrt{x^2+4}dx\\2.&\int\frac{2x+3}{x^2-5x+6}dx\\3.&\int x\log xdx\\4.&\int e^x\left ( \frac{1}{x}-\frac{1}{x^2} \right )dx\\5.&\int \frac{dx}{\sin x+\cos x} \end{align*}\)

(c)

\(\begin{align*}1.&\int \sqrt{\tan x}dx\\2.&\int\frac{1}{\sqrt{x^2-5x+6}}dx\\3.&\int x^2e^xdx \end{align*}\)

Step-by-step solution

Part (a) Step 1: List five integrals that can solve by using the substitution method

\(\begin{align*}1.&\int x^7(x^8-5)^4dx\\2.&\int\frac{e^xdx}{e^x+7}\\3.&\int \frac{(\log x)^5dx}{x}\\4.&\int\frac{\tan^{-1}xdx}{1+x^2}\\5.&\int \frac{dx}{x(x^3-1)} \end{align*}\)

Part (b) Step 2: List five integrals that can not be solved by substitution method

\(\begin{align*}1.&\int \sqrt{x^2+4}dx\\2.&\int\frac{2x+3}{x^2-5x+6}dx\\3.&\int x\log xdx\\4.&\int e^x\left ( \frac{1}{x}-\frac{1}{x^2} \right )dx\\5.&\int \frac{dx}{\sin x+\cos x} \end{align*}\)

Part (c) Step 3: List three integrals that different from part (a) and part (b)

\(\begin{align*}1.&\int \sqrt{\tan x}dx\\2.&\int\frac{1}{\sqrt{x^2-5x+6}}dx\\3.&\int x^2e^xdx \end{align*}\)