Integration strategies

Evaluate the following integrals using …

• substitution
• integration by parts
• trigonometric substitution
• partial fractions

Note: You may check your chosen strategy with the instructor before evaluating.

1. \( \displaystyle\int \frac{1}{\sqrt{10 - x^2}} \,dx \)

hint

2. \( \displaystyle\int \frac{1}{x (x-1)^2} \,dx \)

hint 1

hint 2

3. \( \displaystyle\int x^3 \ln x \,dx \)

hint

5. \( \displaystyle\int \frac{x^3}{x^2 - 1} \,dx \)

hint 1

hint 2

hint 3

3. \( \displaystyle\int \dfrac{1}{\sqrt{x} + 1} \,dx \)

hint

alternate

Answers (mixed up)

• \( \frac{x^2}{2} + \frac{1}{2}\ln|x+1| + \frac{1}{2}\ln|x-1| + C \)
• \( 2\sqrt{x} - 2 \ln(\sqrt{x} + 1) + C \)
• \( \sin^{-1} \left(\frac{x}{\sqrt{10}}\right) + C \)
• \( \ln|x| - \ln|x-1| - \dfrac{1}{x-1} + C\)
• \( \dfrac{x^4 \ln x}{4} - \dfrac{x^4}{16} + C \)