Transcript 1 + cos2x
Evaluating Trigonometric Integrals To evaluate integrals such as ∫sinmx cosnx dx and ∫secmx tannx dx Think ∫u du Evaluating Trigonometric Integrals Think ∫u du ∫sin5x cos4x dx would be? Evaluating Trigonometric Integrals Think ∫u du ∫sin5x cosx dx would be? u du sin6x + c 6 So if we have ∫sin5x cos4x dx what could be do? So if we have ∫sin5x cos4x dx what could be do? What about trying an identity? sin2x + cos2x = 1 sin2x = 1 - cos2x 2 cos2x = 1 + cos2x 2 Guidelines for sine and cosine 1. If sine is odd and positive save 1 sine and convert all others to cosine ∫sin5x cos4x dx = ∫sin x(sin2x)2 cos4x dx ∫sin x(1 - cos2x)2 cos4x dx ∫sin x(1 - 2cos2x + cos4x) cos4x dx ∫cos4x sin x - 2cos6x sin x + cos8x sin x dx Example ∫sin5x cos4x dx = ∫sin x(sin2x)2 cos4x dx ∫sin x(1 - cos2x)2 cos4x dx ∫sin x(1 - 2cos2x + cos4x) cos4x dx ∫(cos4x sin x - 2cos6x sin x + cos8x sin x)dx -∫cos4x -sin x dx+ ∫2cos6x -sin x dx -∫cos8x -sin x dx -1/5 cos5x + 2/7 cos7x - 1/9 cos9x + C Guidelines for sine and cosine 1. If sine is odd and positive save 1 sine and convert all others to cosine 2. If cosine is odd and positive save 1 cosine and convert all others to sine 3. If both are even and positive use the identities sin2x = 1 - cos2x 2 cos2x = 1 + cos2x 2 repeatedly to convert to odd powers of cosine Guidelines for secant & tangent Identity sec2x = 1 + tan2x 1. If secant is even and positive save a sec2x and convert others to tangents 2. If tangent is odd and positive save a secx tanx and convert others to secants 3. If tangent is even and positive & there are no sec x change a tan2x to sec2x 4. If secant is odd and positive & there are no tan xuse integration by parts 5. If nothing else works try converting to sin and cos Example ∫sec43x tan33x dx = ∫(sec23x (sec23x) tan33x)dx ∫(sec23x (1 + tan23x) tan33x)dx ∫tan33x sec23x dx + ∫tan53x sec23x dx 1/3∫tan33x sec23x 3dx + 1/3∫tan53x sec23x 3dx 1/12 tan43x + 1/18 tan63x + C