Calculus

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

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

What do you substitute first? I do not understand what they are asking for. Please help. thanks!

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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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