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"

 

[Soroban]

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
Hence complete answer is 1/4.2/3(1+2x^2)^3/2=1/6(1+2x^2)^3/2

 

[bobmarly12345]

Thankyou very much guys much appreciated