What is the integral of 3ax(ax^2+a^2)^{1/2} ?

The integral of 3ax(ax^2+a^2)^{1/2} is (ax^2+a^2)^{3/2}+C

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integral of 3ax(ax^2+a^2)^{1/2}

\int 3 a x (a x^{2} + a^{2})^{\frac{1}{2}} d x

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

Solution steps

\int 3 a x (a x^{2} + a^{2})^{\frac{1}{2}} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{\frac{1}{2}}}{2 a} d u

= 3 a \cdot \int \frac{u ^{\frac{1}{2}}}{2 a} d u

Take the constant out:

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

= 3 a \frac{1}{2 a} \cdot \int u^{\frac{1}{2}} d u

Apply the Power Rule

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

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

Substitute back

u = a x^{2} + a^{2}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of 3ax(ax^2+a^2)^{1/2} ?
The integral of 3ax(ax^2+a^2)^{1/2} is (ax^2+a^2)^{3/2}+C