What is the integral of kxsqrt(kx^2+10) ?

The integral of kxsqrt(kx^2+10) is 1/3 (kx^2+10)^{3/2}+C

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integral of kxsqrt(kx^2+10)

\int k x k x^{2} + 10 d x

\frac{1}{3} (k x^{2} + 10)^{\frac{3}{2}} + C

Solution steps

\int k x k x^{2} + 10 d x

Take the constant out:

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

= k \cdot \int x k x^{2} + 10 d x

Apply u-substitution

: \int \frac{u}{2 k} d u

= k \cdot \int \frac{u}{2 k} d u

Take the constant out:

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

= k \frac{1}{2 k} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= k \frac{1}{2 k} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = k x^{2} + 10

= k \frac{1}{2 k} \cdot \frac{2}{3} (k x^{2} + 10)^{\frac{3}{2}}

Simplify

k \frac{1}{2 k} \cdot \frac{2}{3} (k x^{2} + 10)^{\frac{3}{2}} : \frac{1}{3} (k x^{2} + 10)^{\frac{3}{2}}

= \frac{1}{3} (k x^{2} + 10)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (k x^{2} + 10)^{\frac{3}{2}} + C

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What is the integral of kxsqrt(kx^2+10) ?
The integral of kxsqrt(kx^2+10) is 1/3 (kx^2+10)^{3/2}+C