Integration by parts

Evaluate the following integrals.

1. \( \displaystyle\int (2x+1) e^x\,dx \)

hint

2. \( \displaystyle\int x \sin(x)\,dx \)

hint

3. \( \displaystyle\int \dfrac{\ln(x)}{x^2}\,dx \)

hint

Challenge problems

C1. \( \displaystyle\int \dfrac{\ln(\ln(x))}{x}\,dx \)

hint 1

alternate hint 1

C2. \( \displaystyle\int x^2 e^x\,dx \)

hint 1

hint 2

C3. \( \displaystyle\int \sin( \sqrt{x}) \,dx \)

hint 1

hint 2

Answers (mixed up)

• \( -2 \sqrt{x} \cos(\sqrt{x}) + 2 \sin(\sqrt{x}) + C\)
• \( x^2 e^x - 2x e^x + 2 e^x + C \)
• \( 2x e^x - e^x + C \)
• \( -\dfrac{1 + \ln(x)}{x} + C \)
• \( \ln(x) \cdot \ln(\ln(x)) - \ln(x) + C \)
• \(\sin(x) - x \cos(x) + C \)