Integral Evaluation and Substitution Methods

What is the purpose of using u-substitution in integration problems?

To simplify the integral by changing the variable to make it easier to evaluate.

Evaluate the integral: ( \int \frac{8x}{(x^2 + 5)^4} \, dx ) using u-substitution.

Let ( u = x^2 + 5 ), then the integral becomes easier to solve.

What is the integral ( \int (3 \, \cos(3x) + 7) \, e^{\sin(3x)} \, dx ) likely simplified to?

Use substitution for ( u = \sin(3x) ) to facilitate integration.

How do you simplify the integral ( \int \frac{\sec^2(4x)}{1 + \tan^2(4x)} \, dx )?

Recognize that ( \sec^2(4x) = 1 + \tan^2(4x) ), simplifying the integral to 1.

What is the integral ( \int \frac{1}{\cos(60)(2 - \sin(60))^7} \, dx ) evaluating at?

Evaluate constants and integrate accordingly.

Evaluate the integral ( \int (2x^2 + 9) \, dx ).

Use standard polynomial integration rules.

What does the integral ( \int \sin^3(t) \cos(t) \, dt ) suggest for substitution?

Let ( u = \sin(t) ) to change the variable.

What is evaluated in ( \int (3x + 1)^5 \, dx )?