Calculus extra credit problem using the rules only of substitution by parts and substitution?
Albert asked:
Hi I have an extra credit problem in my level one calculus class. I need help with it and we can only use two methods substitution by parts and substitution. The problem is x^3(x^2-1)^10. We have to find the integral of this problem. Can anyone help me by only using these two methods showing all work? Thanks in Advance.
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Hi I have an extra credit problem in my level one calculus class. I need help with it and we can only use two methods substitution by parts and substitution. The problem is x^3(x^2-1)^10. We have to find the integral of this problem. Can anyone help me by only using these two methods showing all work? Thanks in Advance.
July 19th, 2010 at 7:30 pm
Hi. I assume when you say subsitution by parts you are obviously referring to what my AP Calculus BC class calls integration by parts which is in the form of uv - ∫v du.
For your problem –> ∫ x^3(x^2-1)^10
Set u = x^2 where du = 2x dx
And dv = x (x^2-1)^10 where you integrate to get v = [½ (x^2-1)^11]/11 or v = [(x^2-1)^11]/22
*Notice one power of x from the term x^3 of the original equation is separated and placed in the expression dv rather than u
Now set up your integration by parts as uv - ∫v du where:
x^2 [(x^2-1)^11]/22 - ∫ [(x^2-1)^11]/22 * 2xdx (Integrate the second expression now by using u = x^2 and du = 2xdx)
So ∫ x^3(x^2-1)10 = x^2 [(x^2-1)^11]/22 - [(x^2-1)^12]/264 jdrawz4