Chapter 5 | Integration
603
⌠ ⌡
−1
⎛ ⎝ 2 x 3 +3 x ⎞ ⎠
⎛ ⎝ x 4 +3 x 2 ⎞ ⎠
2 ∫
u −1 du .
dx = 1
Then we have
1 2 ∫
u −1 du = 1
u | + C
2 ln|
= 1 2 ln |
x 4 +3 x 2 | + C .
Example 5.47 Finding an Antiderivative of a Logarithmic Function
Find the antiderivative of the log function log 2 x .
Solution Follow the format in the formula listed in the rule on integration formulas involving logarithmic functions. Based on this format, we have ∫ log 2 xdx = x ln2 (ln x −1)+ C .
Find the antiderivative of log 3 x .
5.39
Example 5.48 is a definite integral of a trigonometric function. With trigonometric functions, we often have to apply a trigonometric property or an identity before we can move forward. Finding the right form of the integrand is usually the key to a smooth integration. Example 5.48 Evaluating a Definite Integral
Find the definite integral of ⌠ ⌡ 0 π /2
sin x 1+cos x
dx .
Solution We need substitution to evaluate this problem. Let u =1+cos x , , so du =−sin xdx . Rewrite the integral in terms of u , changing the limits of integration as well. Thus, u =1+cos(0)=2 u =1+cos ⎛ ⎝ π 2 ⎞ ⎠ =1.
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