Question

Find three integrals in Exercises 27–70 for which either algebra or u-substitution is a better strategy than integration by parts.

Short answer

$\int x\sin x^{2}dx$, $\int 3x{e}^{x^2}dx$,$\int \frac{3x+1}{x}dx$

Step-by-step solution

Step 1. Given information

Integrals should be easily solved by u- substitution or algebra instead of integration by parts.

Step 2. Observing integrals

• The integral $\int x\sin x^{2}dx$can easily be solved by using u- substitution as for $u=x^2$, the derivative du of u equals $2xdx$. Hence, the transformed integral $\frac{1}{2}\int \sin udu$can be solved simply.
• The integral $\int 3x{e}^{x^2}dx$can easily be solved by using u- substitution as for $u=x^2$, the derivative du of u equals $2xdx$. Hence, the transformed integral $3\int \frac{e^u}{2}du$can be solved simply.
• The integral $\int \frac{3x+1}{x}dx$can easily be solved by using algebra because the integrand can be simplified as$3+\frac{1}{x}$. Hence, the transformed integral $\int 3+\frac{1}{x}dx$can be solved simply.