calculus: evaluate int[(x^9)cos(x^5)dx

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thebenji

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How to evaluate the indefinite integral:

. . .int[(x^9)cos(x^5)dx

I couldn't think of any possible u substitutions and substituting for trig functions didn't work for me either. Any help?
 
One way to see how ‘they got it’ is the take the answer, differentiate it, and work with it until you get back to where you began.
 
thebenji said:
How to evaluate the indefinite integral:

. . .int[(x^9)cos(x^5)dx

I couldn't think of any possible u substitutions and substituting for trig functions didn't work for me either. Any help?
Click to expand...

u = x^5
du = 5x^4 dx

INT x^9*cos(x^5) dx =
(1/5)INT 5x^4*x^5*cos(x^5) dx =
(1/5)INT u*cos(u) du

integration by parts performed by tabular integration ...

+ ... u ... cos(u)
- ... 1 ... sin(u)
+ ... 0 ... -cos(u)

(1/5)INT u*cos(u) du = (1/5)[u*sin(u) + cos(u)] + C

back substitute x^5 for u and you're there.
 
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