intergrate with respect to x:
x√1+2x² dx
need the steps how to solve please
thank you
Hello, bobmarly12345!
$\displaystyle \text{Integrate: }\:\int x \sqrt{1+2x^2}\,dx$
We have: .$\displaystyle \int (1 + 2x^2)^{\frac{1}{2}}(x\,dx)$
Let: .$\displaystyle u \,=\,1+2x^2 \quad\Rightarrow\quad du \,=\,4x\,dx \quad\Rightarrow\quad x\,dx \,=\,\tfrac{1}{4}du$
Substitute: .$\displaystyle \int \!u^{\frac{1}{2}}\left(\tfrac{1}{4}du\right) \;=\;\tfrac{1}{4} \!\!\int \!u^{\frac{1}{2}}du \;=\;\tfrac{1}{6}u^{\frac{3}{2}} + C$
Back-substitute: .$\displaystyle \tfrac{1}{6}(1 + 2x^2)^{\frac{3}{2}} + C$
Write question as 1/4 Integral of 4x(1+2x^2)^1/2
There is a rule which says as 4x is the derivative of the expression in the bracket then this part integrates to (1+2x^2)^3/2 divided by 3/2, that is
2/3(1+2x^2)^3/2
Hence complete answer is 1/4.2/3(1+2x^2)^3/2=1/6(1+2x^2)^3/2