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How do you find the antiderivative of #cos(x) Ln(sin(x)) #?

Answer 1

Scarlett Allison

Use substitution and parts.

#I = int cosxln(sinx) dx#

Let #t = sinx# so that #dt = cosx dx# and

#I = int lnt dt#

This can be integrated using parts

# u = lnt# so #du = 1/t dt# #dv = dt#, so #v = t#

#I = tlnt-int t * 1/t dt#

# = tlnt- int dt#

# = tlnt-t+C#

Reverse the substitution to finish with

#int cosxln(sinx) dx = sinxln(sinx) - sinx+C#

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Answer 2

Ezra Adams

To find the antiderivative of cos(x) ln(sin(x)), you can use integration by parts. Let u = ln(sin(x)) and dv = cos(x) dx. Then differentiate u and integrate dv to find du and v respectively. Apply the integration by parts formula: ∫u dv = uv - ∫v du. Finally, substitute the values of u, v, dv, and du into the integration by parts formula and evaluate the integral.

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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.