CPM Homework Help : CALC Problem 9-64

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9-64.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy. After evaluating each integral, write a short description of your method.

$\int \frac { 4 x } { \sqrt { 1 + x ^ { 2 } } } d x$
Step (a):
Use substitution: Let $u = x^2 + 1$

$\int \operatorname { cos } ( 4 \theta ) \operatorname { sin } ( 2 \theta ) d \theta$
Hint (b):
$\operatorname{cos}(4θ) =\operatorname{cos}(2(2θ)) = 2(\operatorname{cos}^2(2θ)) − 1$
Step (b):
Substitute using the hint, then integrate using substitution.
Let $u = \operatorname{cos}(2θ)$ .

$\int \frac { 4 } { \sqrt { 1 - x ^ { 2 } } } d x$
Hint (c):
The integrand is a multiple of an inverse trigonometric function derivative.

$\int 6 x ^ { 3 } \cdot 2 ^ { x ^ { 4 } } d x$
Step (d):
Use substitution: Let $u = x^4$

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