Lecture Notes: Integration by Parts and u-Substitution Examples

Example 1: Integral of $e^{2x} \cdot \sin(3x)$

Integration by Parts Setup

• Choose $u = e^{2x}$ and $dv = \sin(3x)dx$.
  • Derivative: $du = 2e^{2x}dx$.
  • Integral: $v = -\frac{1}{3} \cos(3x)$.
• Integration by parts formula: $\int u , dv = uv - \int v , du$.

First Step

• $uv = -\frac{1}{3} e^{2x} \cos(3x)$.
• $\int v , du = -\frac{2}{3} \int e^{2x} \sin(3x) dx$.
• The integral does not become simpler; use integration by parts again.

Second Integration by Parts

• Choose again $u = e^{2x}$ and $dv = \cos(3x)dx$.
  • $du = 2e^{2x}dx$.
  • $v = \frac{1}{3} \sin(3x)$.
• Apply integration by parts: $\frac{1}{3} e^{2x} \sin(3x) - \frac{2}{3} \int e^{2x} \sin(3x) dx$.

Solving the Integral

• Recognize the integral forms a cycle and solve by algebraic manipulation.
• Final result after solving algebraically:
  • $\int e^{2x} \sin(3x) dx = \frac{9}{13} \left(-\frac{1}{3} e^{2x} \cos(3x) + \frac{2}{3} \left(\frac{1}{3} e^{2x} \sin(3x)\right)\right) + C$.

Example 2: Integral of $\text{arcsin}(3t)$

Setup and Derivatives

• Choose $u = \text{arcsin}(3t)$, $dv = dt$.
  • Derivative: $du = \frac{3}{\sqrt{1-(3t)^2}} dt$.
  • $v = t$.
• Integration by parts formula: $\int u , dv = uv - \int v , du$.

Integration by Parts

• $uv = t \cdot \text{arcsin}(3t)$.
• $\int v , du = \int \frac{3t}{\sqrt{1-9t^2}} dt$.

u-Substitution for Remaining Integral

• Use $u$-substitution: $u = 1 - 9t^2$.
  • $du = -18t , dt$.
  • $6 , du = -3t , dt$.
• Substitute and integrate: $\int \frac{1}{\sqrt{u}} du = 2\sqrt{u} + C$.

Final Result

• Substitute back $u = 1 - 9t^2$.
• Combine results: $t \cdot \text{arcsin}(3t) + \frac{1}{3} \cdot 2\sqrt{1-9t^2} + C$.