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

The integral of 2x*sqrt(1+x^2) is 2/3 (1+x^2)^{3/2}+C

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

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

\frac{2}{3} (1 + x^{2})^{\frac{3}{2}} + C

Solution steps

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

Take the constant out:

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

= 2 \cdot \int x 1 + x^{2} d x

Apply u-substitution

= 2 \cdot \int u^{2} d u

Apply the Power Rule

= 2 \cdot \frac{u ^{3}}{3}

Substitute back

u = 1 + x^{2}

= 2 \cdot \frac{( 1 + x ^{2} ) ^{3}}{3}

Simplify

2 \cdot \frac{( 1 + x ^{2} ) ^{3}}{3} : \frac{2}{3} (1 + x^{2})^{\frac{3}{2}}

= \frac{2}{3} (1 + x^{2})^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (1 + x^{2})^{\frac{3}{2}} + C

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What is the integral of 2x*sqrt(1+x^2) ?
The integral of 2x*sqrt(1+x^2) is 2/3 (1+x^2)^{3/2}+C