How do I evaluate the following integral by parts? #intx^9cos(x^5)dx#

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#intx^9cos(x^5)dx#

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Answer 1 · 2018-01-18T23:49:09

# I = (x^5sin (x^5) + cos (x^5))/5 + C #

We seek:

# I = int \ x^9 \ cos(x^5) \ dx #

We should first reduce the term #x^5# in the integral by performing a substitution:

Let # \ t =x^5 => (dt)/dx =5x^4 #

So we can manipulate the integral, and perform this substitution to get

# I = int \ 1/5 \ x^5 \ cos(x^5) \ (5x^4) \ dx #
# \ \ = 1/5 \ int \ t \ cos t \ du #

And now we can apply Integration By Parts:

Let # { (u,=t, => (du)/dt,=1), ((dv)/dt,=cost, => v,=sint ) :}#

Then plugging into the IBP formula:

# int \ u \ dv = uv - int v \ du #

We have:

# int \ t \ cost \ dt = (t)(sint) -int \ (sint)(1) \ dt #
# " " = tsin t + cos t \ \ (+c) #

Allowing us to write:

# I = 1/5 {tsin t + cos t} + C #

Then restoring the substitution, we get:

# I = (x^5sin (x^5) + cos (x^5))/5 + C #