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7.2 Notes

# 7.2 Notes - x to replace remaining sec x and use the...

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Trigonometric Integrals (Section 7.2) We’ll start by evaluating: Z tan xdx Z sec xdx Similar techniques can be used to find: Z cot xdx = ln | sin x | + C Z csc xdx = - ln | csc x + cot x | + C Integrals of the form Z sin m x cos n xdx : 1. If n is odd, save one factor of cos x and use cos 2 x = 1 - sin 2 x along with the substitution u = sin x . Example 0.1. Z sin 2 x cos 3 xdx

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2. If m is odd, save a factor of sin x and use sin 2 x = 1 - cos 2 x along with the substitution u = cos x . Example 0.2. Z sin 5 x cos 4 xdx 3. If m and n are both odd, use either of the above methods. 4. If both powers are even use sin 2 x = 1 2 (1 - cos 2 x ) and cos 2 x = 1 2 (1 + cos 2 x ). Example 0.3. Z π/ 2 0 sin 2 2 xdx Integrals of the form Z tan m x sec n xdx : 1. If n is even, save a factor of sec 2 x , use sec 2 x = 1 + tan

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Unformatted text preview: x to replace remaining sec x and use the substitution u = tan x . Example 0.4. Z tan 2 x sec 4 xdx 2. If m is odd, save a sec x tan x and use tan 2 x = sec 2 x-1 and use the substitution u = sec x . Example 0.5. Z tan 3 sec xdx Example 0.6. Z tan 4 dx 3. When neither of the above work, use integration by parts to reduce exponents. Example 0.7. Z sec 3 xdx Sometimes it works nicely to convert to sin x and cos x . Example 0.8. Z csc ± t 2 ² sin 3 ± t 2 ² dt Example 0.9. Z cot 2 (4 x ) sin 4 (4 x ) dx Example 0.10. Z (sec 2 x-1) dx...
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7.2 Notes - x to replace remaining sec x and use the...

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