What is the integral of 5xsqrt(5x^2-12) ?

The integral of 5xsqrt(5x^2-12) is 1/3 (5x^2-12)^{3/2}+C

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integral of 5xsqrt(5x^2-12)

\int 5 x 5 x^{2} - 12 d x

\frac{1}{3} (5 x^{2} - 12)^{\frac{3}{2}} + C

Solution steps

\int 5 x 5 x^{2} - 12 d x

Take the constant out:

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

= 5 \cdot \int x 5 x^{2} - 12 d x

Apply u-substitution

: \int \frac{u}{10} d u

= 5 \cdot \int \frac{u}{10} d u

Take the constant out:

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

= 5 \cdot \frac{1}{10} \cdot \int u d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{10} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 5 x^{2} - 12

= 5 \cdot \frac{1}{10} \cdot \frac{2}{3} (5 x^{2} - 12)^{\frac{3}{2}}

Simplify

5 \cdot \frac{1}{10} \cdot \frac{2}{3} (5 x^{2} - 12)^{\frac{3}{2}} : \frac{1}{3} (5 x^{2} - 12)^{\frac{3}{2}}

= \frac{1}{3} (5 x^{2} - 12)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{3} (5 x^{2} - 12)^{\frac{3}{2}} + C

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What is the integral of 5xsqrt(5x^2-12) ?
The integral of 5xsqrt(5x^2-12) is 1/3 (5x^2-12)^{3/2}+C