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What is the integral of #int sin^5x*cos^7x #?

Answer 1

#-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C#

# sin^5x*cos^7x = sin^4x cos^7x sinx = (1-cos^2 x)^2cos^7x sin x# #=(1-2cos^2x+cos^4 x)cos^7 x sin x # #=(cos^7x-2 cos^9 x+cos^11 x)sinx# so

#int sin^5x*cos^7x dx = int (cos^7x-2 cos^9 x+cos^11 x)sinx dx# #=-(1/8cos^8 x-2/10 cos^10 x + 1/12 cos^12 x) + C#

or simplifying

#-1/480(27 - 28 cos(2 x) + 5 cos(4 x)) cos^8x#

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Answer 2

Charles Anctil

To find the integral of ( \int \sin^5(x) \cos^7(x) , dx ), you can use trigonometric identities to rewrite the integral in terms of powers of sine and cosine. One approach is to use substitution, where you let ( u = \sin(x) ) or ( u = \cos(x) ) and then express ( \sin^5(x) ) or ( \cos^7(x) ) in terms of ( u ). Another approach is to use integration by parts, where you split the integrand into two functions and apply the integration by parts formula. Both methods should lead to a solution, but the specific choice depends on personal preference and ease of computation.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.