Flashcards in 9.3.1 Integrating Composite Trigonometric Functions by Substitution Deck (12):
Integrating Composite Trigonometric Functions by Substitution
• Integration by substitution is a technique for finding the antiderivative of a composite function. A composite function is a function that results from first applying one function, then another.
• If the du-expression is only off by a constant multiple, you can still use integration by substitution by moving that constant out of the integral.
- This integral involves a composite function: the sine of a complicated expression. If you let u be the inside of the function, notice that du is found surrounding the sine function.
- After you substitute u, make sure that nothing remains in terms of x.
- Recall that the derivative of –cosx is sinx.
- Make sure to replace u with its expression in terms of x.
- You can check that your answer is correct by taking its
- Here is another composite function. Let u be the inside
expression. When you find du, you will notice that there is no multiple of 4 in the integrand, just dx.
- Since 4 is just a constant multiple, solve for dx and substitute that expression into the integrand.
- You can move 1/4 outside the integrand since it is a constant multiple.
- After you integrate, make sure to replace u with its expression in terms of x.
- Take the derivative of your answer to make sure it is correct.
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