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

1. ## 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.

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

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 $\displaystyle dv=e^{4t}$

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

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

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

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

4. ## 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.

5. ## 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.

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

I substituted $\displaystyle x=t^2$

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

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

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

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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