# Calculus

First make a substitution and then use integration by parts to evaluate.

The integral of (x^9)(cos(x^5))dx

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1. let integral (x^9)(cos(x^5))dx
= integral (1/5)x^5( (5x^4)cos(x^5))dx

let dv= (1/5)x^4cos(x^5)dx , let u = (1/5)x^5
v = sin(x^5) , du = x^4 dx

so integral of (x^9)(cos(x^5))dx
= uv - integral [vdu
= (1/5)x^5(sin(x^5) - integral [ x^4(sin(x^5))dx
= (1/5)x^5(sin(x^5) - (1/5)(-cos(x^5)
= (1/5)[ (x^5(sin(x^5)) + cos(x^5) ]

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