Calculus/Integration techniques/Trigonometric Integrals

Integration techniques/Trigonometric Substitution Integration techniques/Partial Fraction Decomposition When the integrand is primarily or exclusively based on trigonometric functions, the following techniques are useful. Powers of Sine and Cosine We will give a general method to solve generally integrands of the form . First let us work through an example. Notice that the integrand contains an odd power of cos. So rewrite it as We can solve this by making the substitution so . Then we can write the whole integrand in terms of by using the identity . So {| | | |- | | |- | | |- | | |- | | |} This method works whenever there is an odd power of sine or cosine. To evaluate when either or is odd. If is odd substitute and use the identity . If is odd substitute and use the identity . Example Find . As there is an odd power of we let so . Notice that when we have and when we have . {| | | |- | | |- | | |- | | |- | | |- | | |} When both and are even, things get a little more complicated. To evaluate when both and are even. Use the identities and . Example Find . As and we have and expanding, the integrand becomes Using the multiple angle identities {| | | |- | | |- | |TODO: CORRECT FORMULA |} then we obtain on evaluating Powers of Tan and Secant To evaluate . If is even and then substitute and use the identity . If and are both odd then substitute and use the identity . If is odd and is even then use the identity and apply a reduction formula to integrate , using the examples below to integrate when . Example 1 Find . There is an even power of . Substituting gives so Example 2 Find . Let so . Then {| | | |- | | |- | | |- | | |- | | |} Example 3 Find . The trick to do this is to multiply and divide by the same thing like this: {| | | |- | | |} Making the substitution so , {| | | |- | | |- | | |} More trigonometric combinations For the integrals or or use the identities Example 1 Find . We can use the fact that , so Now use the oddness property of to simplify And now we can integrate {| | | |- | | |} Example 2 Find: . Using the identities Then {| | | |- | | |} Integration techniques/Trigonometric Substitution Integration techniques/Partial Fraction Decomposition