Question #a2f4c

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Answer 1 · 2017-10-05T14:55:43

This list of integrals contains a reduction formula for

$intsin^n(x)dx$

we shall recursively use it.

The reduction formula is:

$intsin^n(x)dx=(sin^(n-1)(x)cos(x))/n+(n-1)/nintsin^(n-2)(x)dx$

Substituting 6 into the formula:

$intsin^6(x)dx=(sin^5(x)cos(x))/6+5/6intsin^4(x)dx$

Substituting 4 into the formula:

$intsin^6(x)dx=(sin^5(x)cos(x))/6+5/6[(sin^3(x)cos(x))/4+3/4intsin^2(x)dx]$

Substituting 2 into the formula:

$intsin^6(x)dx=(sin^5(x)cos(x))/6+5/6[(sin^3(x)cos(x))/4+3/4{(sin(x)cos(x))/2+1/2intdx}]$

We know the last integral:

$intsin^6(x)dx=(sin^5(x)cos(x))/6+5/6[(sin^3(x)cos(x))/4+3/4{(sin(x)cos(x))/2+1/2x}] + C$