What is the integral of x/(sqrt(1+2x^2)) ?

The integral of x/(sqrt(1+2x^2)) is 1/2 sqrt(1+2x^2)+C

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integral of x/(sqrt(1+2x^2))

\int \frac{x}{1 + 2 x ^{2}} d x

\frac{1}{2} 1 + 2 x^{2} + C

Solution steps

\int \frac{x}{1 + 2 x ^{2}} d x

Apply u-substitution

: \int \frac{1}{4 u} d u

= \int \frac{1}{4 u} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{4} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 1 + 2 x^{2}

= \frac{1}{4} \cdot 2 (1 + 2 x^{2})^{\frac{1}{2}}

Simplify

\frac{1}{4} \cdot 2 (1 + 2 x^{2})^{\frac{1}{2}} : \frac{1}{2} 1 + 2 x^{2}

= \frac{1}{2} 1 + 2 x^{2}

Add a constant to the solution

= \frac{1}{2} 1 + 2 x^{2} + C

\frac{d y}{d x} + y (1 - 2 x) = - \frac{x ^{2}}{2} y

\int 50 + 3 x^{2} - \frac{3}{2} x d x

\int \frac{8 x ^{2}}{4 x ^{2} + 7} d x

\frac{d}{d x} (- \frac{4}{( 2 x + 1 ) ^{2}})

e^{s^{2}}

What is the integral of x/(sqrt(1+2x^2)) ?
The integral of x/(sqrt(1+2x^2)) is 1/2 sqrt(1+2x^2)+C