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Chapter 4 | Applications of Derivatives
if and only if F is an antiderivative of f . Therefore, when claiming that ∫ f ( x ) dx = F ( x )+ C it is important to check whether this statement is correct by verifying that F ′( x ) = f ( x ).
Example 4.51 Verifying an Indefinite Integral Each of the following statements is of the form ∫ f ( x ) dx = F ( x )+ C . Verify that each statement is correct by showing that F ′( x ) = f ( x ). a. ∫ ( x + e x ) dx = x 2 2 + e x + C b. ∫ xe x dx = xe x − e x + C
Solution
a. Since
⎛ ⎝ x 2
⎞ ⎠ = x + e x ,
d dx
e x + C
2 +
the statement
∫ ( x + e x ) dx = x 2
e x + C
2 +
is correct. Note that we are verifying an indefinite integral for a sum. Furthermore, x 2 2 and e x are antiderivatives of x and e x , respectively, and the sum of the antiderivatives is an antiderivative of the sum. We discuss this fact again later in this section. b. Using the product rule, we see that d dx ( xe x − e x + C ) = e x + xe x − e x = xe x .
Therefore, the statement
∫ xe x dx = xe x − e x + C
is correct. Note that we are verifying an indefinite integral for a product. The antiderivative xe x − e x is not a product of the antiderivatives. Furthermore, the product of antiderivatives, x 2 e x /2 is not an antiderivative of xe x since d dx ⎛ ⎝ x 2 e x 2 ⎞ ⎠ = xe x + x 2 e x 2 ≠ xe x .
In general, the product of antiderivatives is not an antiderivative of a product.
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