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Chapter 5 | Integration
Example 5.38 Square Root of an Exponential Function
Find the antiderivative of the exponential function e x 1+ e x .
Solution First rewrite the problem using a rational exponent:
∫ e x 1+ e x dx = ∫ e x (1+ e x ) 1/2 dx . Using substitution, choose u =1+ e x . u =1+ e x . Then, du = e x dx . We have ( Figure 5.37 ) ∫ e x (1+ e x ) 1/2 dx = ∫ u 1/2 du . Then ⌠ ⌡ u 1/2 du = u 3/2 3/2 + C = 2 3 u 3/2 + C = 2 3 (1+ e x ) 3/2 + C .
Figure 5.37 The graph shows an exponential function times the square root of an exponential function.
Find the antiderivative of e x (3 e x −2) 2 .
5.32
Example 5.39 Using Substitution with an Exponential Function
Use substitution to evaluate the indefinite integral ∫ 3 x 2 e 2 x 3 dx .
Solution Here we choose to let u equal the expression in the exponent on e . Let u =2 x 3 and du =6 x 2 dx .. Again, du is off by a constant multiplier; the original function contains a factor of 3 x 2 , not 6 x 2 . Multiply both sides of the equation by 1 2 so that the integrand in u equals the integrand in x . Thus, ∫ 3 x 2 e 2 x 3 dx = 1 2 ∫ e u du .
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