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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