# intergrate with respect to x

#### bobmarly12345

intergrate with respect to x:

x√1+2x² dx

need the steps how to solve please

thank you

#### princeps

intergrate with respect to x:

x√1+2x² dx

need the steps how to solve please

thank you
WA click "show steps"

Last edited:

#### Soroban

MHF Hall of Honor
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: .$$\displaystyle \tfrac{1}{6}(1 + 2x^2)^{\frac{3}{2}} + C$$

#### biffboy

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