Integration Methods Problems

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Below are some example problems. Please click the [show] link to view the solutions.

Example 1:

Question 1
∫30 ${\frac {dx}{\sqrt {4-x}}}$

Show Solution
Use integration by substitution. u = 4 - x ${\frac {du}{-1}}$ = dx ∫14-u^${\frac {-1}{2}}$ = 2u^(${\frac {1}{2}}$) |14 = -2 - (-4) = 2

Example 2:

Question 2
∫10 x$e^{x}$dx

Show Solution
Use integration by parts. u = x ${\frac {du}{dx}}=1$ du = dx dv = $e^{x}$dx v = $e^{x}$ ∫udv = uv - ∫vdu = x$e^{x}$ - ∫$e^{x}$dx = x$e^{x}$ - $e^{x}$ |10 = 0 - (0 - 1) = 1

Example 3:

Question 3
∫10 x$e^{x}$dx

Show Solution
Use integration by parts. u = x ${\frac {du}{dx}}=1$ du = dx dv = $e^{x}$dx v = $e^{x}$ ∫udv = uv - ∫vdu = x$e^{x}$ - ∫$e^{x}$dx = x$e^{x}$ - $e^{x}$ |10 = 0 - (0 - 1) = 1

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