Integration by Parts: x^2(2+x^3)^4, x(x^2+2)^5, xcosx

[pinnacleprouk]

Could you please confirm my answers as correct or incorrect, if incorrect please point out errors and assist in correcting errors.

Thanks in advance

Homework Statement

Intergrate the following functions with respect to x using recognition, substitution or intergration by parts:

x^2(2+x^3)^4

x(x^2+2)^5

xcosx

The Attempt at a Solution

1)
integral(x^2(2+x^3)^4)
put (2+x^3) =x
thus
3x^2dx = dz
or
x^2dx = dz/3

the integral becomes
integral(z^4dz)/3
= z^5/15
=(2+x^3)^5/15 + c

2)
integral(x(x^2+2)^5)
put x^2+2=z
or
2xdx = dz
thus
xdx = dz/2
integral(z^5/2)dz
= z^6/12+c
=(x^2+2)^6/12 + c

3) integral(xcosx)
x(sinx) - integral(sinx)
= -xsinx+cosx +c

 

[Susanne217]

Could you please confirm my answers as correct or incorrect, if incorrect please point out errors and assist in correcting errors.

Thanks in advance

Homework Statement

Intergrate the following functions with respect to x using recognition, substitution or intergration by parts:

x^2(2+x^3)^4

x(x^2+2)^5

xcosx

The Attempt at a Solution

1)
integral(x^2(2+x^3)^4)
put (2+x^3) =x
thus
3x^2dx = dz
or
x^2dx = dz/3

the integral becomes
integral(z^4dz)/3
= z^5/15
=(2+x^3)^5/15 + c

2)
integral(x(x^2+2)^5)
put x^2+2=z
or
2xdx = dz
thus
xdx = dz/2
integral(z^5/2)dz
= z^6/12+c
=(x^2+2)^6/12 + c

3) integral(xcosx)
x(sinx) - integral(sinx)
= -xsinx+cosx +c

You can test your final result using either mathematical software like Maple, Mathematica and or google Wolfram integrator it using the integration engine of Mathematica.

 

[pinnacleprouk]

while I appreciate your reply, I would rather ask here to see if the steps taken are correct as well as the final result!

Thanks

 

[DJsTeLF]

1)
integral(x^2(2+x^3)^4)
put (2+x^3) =x
thus...

I think you mean (2+x^3) = z

Last one is definitely correct and, albeit a little tedious, the other 2 can easily be confirmed by expanding the brackets and integrating directly.

 

[Susanne217]

while I appreciate your reply, I would rather ask here to see if the steps taken are correct as well as the final result!

Thanks

Okay, but as you know which I learned yesterday its forbidden to reply with re-calculations to you.

2) and 3) are correct but you need to redo 1) using integration by parts and integration by substitution!