Integral of t^3*(e^4t^2) and t^2*(e^4t)

Been doing plenty of practice of late just to get back in the swing of things. Could someone please help me out with the solutions to the above?

I think they may involve integration by parts twice, which sounds ugly.

I can't find any similar examples in notes or text book...

Regards.

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

Quote:

Originally Posted by

**paulbk108** Been doing plenty of practice of late just to get back in the swing of things. Could someone please help me out with the solutions to the above?

I think they may involve integration by parts twice, which sounds ugly.

I can't find any similar examples in notes or text book...

Regards.

For the first one take: $\displaystyle u=t^2$ and $\displaystyle dv=t\cdot e^{4t^2} dt$

For the second one take$\displaystyle u=t^2$ and $\displaystyle dv=e^{4t}$

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

Quote:

$\displaystyle \int t^3(e^{4t^2}) \mathrm{d}t$

Substitute $\displaystyle x=t^2$.

$\displaystyle \int t^3(e^{4t^2})\ \mathrm{d}t=\int \frac{x \cdot e^{4x}}{2}\ \mathrm{d}x$

$\displaystyle ={1 \over 2}\left[ \frac{e^{4x}x}{4}-\frac{1}{4}\int e^{4x} dx\right]+C={1 \over 2}\left[ \frac{e^{4x}x}{4}-\frac{1}{16}e^{4x}\right]+C$

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

One of you folks mind completing one of them?

Once I get rolling I will be fine...

Regards.

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

sbhatnagar,

Can you please explain where the t^3 went? You substituted x = t^2...?

Regards.

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

I substituted $\displaystyle x=t^2$

$\displaystyle \frac{dx}{dt}=2t \iff dt=\frac{dx}{2t}$

$\displaystyle \int t^3{e^{4t^2}} dt=\int \frac{t^3(e^{4x})}{2t} dx=\int\frac{t^2(e^{4x})}{2} dx=\int \frac{x \cdot e^{4x}}{2} dx$

Re: Integral of t^3*(e^4t^2) and t^2*(e^4t)

Quote:

Originally Posted by

**paulbk108** One of you folks mind completing one of them?

Once I get rolling I will be fine...

Regards.

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