Home > APCALC > Chapter 9 > Lesson 9.2.2 > Problem9-63

9-63.

Examine the integrals below. Consider the multiple tools available for integrating and use the best strategy for each part. Evaluate each integral and briefly describe your method.

1. $\int \frac { 4 x } { \sqrt { 1 + x ^ { 2 } } } d x$

Use substitution: Let $u = x^2 + 1$

1. $\int \operatorname { cos } ( 4 \theta ) \operatorname { sin } ( 2 \theta ) d \theta$

$\cos(4θ) = \cos(2(2θ)) = 2(\cos^2(2θ)) − 1$

Substitute using the hint, then integrate using substitution.
Let $u = \cos(2θ)$.

1. $\int \frac { 4 } { \sqrt { 1 - x ^ { 2 } } } d x$

The integrand is a multiple of an inverse trigonometric function derivative.

1. $\int 6 x ^ { 3 } \cdot 2 ^ { x ^ { 4 } } d x$

Use substitution: Let $u = x^4$