Evaluate \int \sin^{5}xdx.

e1s2kat26 2021-10-07 Answered
Evaluate sin5xdx.
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Expert Answer

SchulzD
Answered 2021-10-08 Author has 83 answers
Step 1
In some cases integrals can be simplified by use of a substitution. This usually results in simplifying the integral in terms of known standard integrals.
For the given integral the integrand can be written as sin(x)sin4(x)=sin(x)(1cos2(x))2.
Then the substitution u=cos(x) will simplify the integrand expression.
Step 2
The given indefinite integral to be evaluated is sin5(x)dx. This can be rewritten as sin(x)(1cos2(x))2dx.Use the substitution u=cos(x). Differentiating this gives du=sin(x)dx.Substitute this in the integral and evaluate the integral.
sin5(x)dx=sin(x)(1cos2(x))2dx
=(1u2)2du
=(1+u42u2)du
=uu55+2u33+C
=cos(x)cos5(x)5+2cos3(x)3+C
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