• p 418 #8
    1. \( \int x^2 e^{ x^3 } dx \)
    2. let \( u = x^3 \)
    3. \( \frac{du}{dx} = 3 x^2 \)
    4. \( \frac{1}{3} x^{ -2 } du = dx \)
    5. \( \int x^2 e^{ x^3 } \cdot \frac{1}{3} x^{ -2 } du \)
    6. \( \int \frac{1}{3} x^2 x^{ -2} e^u du \)
    7. \( \frac{1}{3} \int e^u du \)
    8. \( \frac{1}{3} e^{x^3} + C \)

      9. \( \int x^2 e^{x^3} dx = \frac{1}{3} e^{ x^3 } + C \)

  • p 418 #10
    1. \( \int \sin(t) \sqrt{1 + \cos(t) } dt \)
    2. let \( u = 1 + \cos(t) \)
    3. \( \frac{du}{dt} = - \sin(t) \)
    4. \( dt = - \frac{1}{\sin(t)} du \)
    5. \( \int \sin(t) \cdot \frac{-1}{\sin(t)} \cdot u^{ \frac{1}{2} } du \)
    6. \( - \int u^{\frac{1}{2}} du\)
    7. \( - \frac{2}{3} u^{ \frac{3}{2} } + C \)
    8. \( - \frac{2}{3}( 1 + \cos(t))^{ \frac{3}{2} } + C\)

      9. \( \int \sin(t) \sqrt{1 + \cos(t) } dt = - \frac{2}{3} ( 1 + \cos(t))^{ \frac{3}{2} } + C \)