Practice: Integration by Substitution

Evaluate the following integrals.

1. $\int 2x \sin (x^2) dx$

2. $\int \frac{x}{(x^2+1)} dx$

3. $\int \frac{1}{x\ln(x)} dx$

4. $\int \cot(\theta) d\theta$ (Hint: $\cot \theta = \frac{\cos \theta}{\sin \theta}$)

5. $\int e^x \sec^2 (e^x) dx$

6. $\int \sqrt{1+x} dx$, using:

   • $u =1+x$;

   • $u=\sqrt{1+x}$.

7. $\int_1^e \ln (x^{1/x})dx$

8. $\int_0^1 x^3 \sqrt{1-x^2}dx$

9. $\int_0^1 x^5 \sqrt{1-x^2}dx$

• Answers

  1. $-\cos (x^2) + C$

  2. $\ln|x^2+1| + C$ (Note: since $x^2+1$ is always positive, it is OK to drop the absolute value sign.)

  3. $\ln | \ln |x| | + C$

  4. $\ln (|\sin x|) + C$

  5. $\tan(e^x) + C$

  6. $\frac{2}{3} (1+x)^{3/2} + C$

  7. $\frac{1}{2}$

  8. $\frac{2}{15}$

  9. $\frac{8}{105}$