Help with Indefinite Integrals??

[lostit]

So I missed a few classes of my intro to calculus class and now I'm lost.

If anyone has the time to catch me back up.

These are some of the problems I'm not sure how to do.
Find the Indefinite Integral:

e^x^3(3x^3)dx

(1/(5x-7))(5)dx

(square root of (1=x^4))(4x^3)dx

If any one can example these to me please do.

thanks

 

[soroban]

Hello, lostit!

Looks like you missed the lessons on Substitution.

From the level of the last two problems,
I assume there is a typo in the first one . . .

$\displaystyle \displaystyle \int e^{x^3}(3x^2\,dx)$

$\displaystyle \text{Let }u \,=\, x^3 \quad\Rightarrow\quad du \,=\, 3x^2\,dx$

Substitute: .$\displaystyle \displaystyle \int e^u\,du \;=\;e^u + C$

Back-substitute: .$\displaystyle e^{x^3} + C$

$\displaystyle \displaystyle\int \frac{5\,dx}{5x-7}$

$\displaystyle \text{Let }u \,=\, 5x-7 \quad\Rightarrow\quad du \,=\,5\,dx$

Substitute: .$\displaystyle \displaystyle\int\frac{du}{u} \;=\;\ln|u| + C$

Back-substitute: .$\displaystyle \ln|5x-7| + C$

$\displaystyle \displaystyle \int \sqrt{1+x^4}\,(4x^3\,dx)$

$\displaystyle \displaystyle\text{We have: }\:\int (1+x^4)^{\frac{1}{2}}(4x^3\,dx)$

$\displaystyle \text{Let }u \,=\,1+x^4 \quad\Rightarrow\quad du \,=\,4x^3\,dx$

Substitute: .$\displaystyle \displaystyle \int u^{\frac{1}{2}}\,du \;=\;\tfrac{2}{3}u^{\frac{3}{2}} + C$

Back-substitute: .$\displaystyle \tfrac{2}{3}(1+x^4)^{\frac{3}{2}}+C$