# First make a substitution and then use integration by parts to evaluate the integral: (x^7)(cos(x^4))dx?

First make a substitution and then use integration by parts to evaluate the integral: (x^7)(cos(x^4))dx

Relevance

∫(x^7)(cos(x^4)dx

Now substitute:

t=x^4

dt/dx=4x^3

dx=dt/4x^3

= ∫(x^7)(cos(t))dt/4x^3

= 1/4∫(x^7)/(x^3)(cos(t))dt

= 1/4∫(x^4)(cos(t))dt

Remember that t=x^4:

= 1/4∫t*cos(t)dt

Integration by parts:

∫u'*v = [u*v] - ∫u*v'

u' = cos(t) => u = sin(t)

v = t => v' = 1

= 1/4 (sin(t)*t - ∫sin(t)*1dt) + C

= 1/4 [sin(t)*t - (-cos(t))] + C

Now substitute back t=x^4

= 1/4 (sin(x^4)*x^4 + cos(x^4))+C

I hope I didn't make any mistakes

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