What is the integral of (24x)/(4sqrt(4x^2+8)) ?

The integral of (24x)/(4sqrt(4x^2+8)) is 3sqrt(x^2+2)+C

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integral of (24x)/(4sqrt(4x^2+8))

\int \frac{24 x}{4 4 x ^{2} + 8} d x

3 x^{2} + 2 + C

Solution steps

\int \frac{24 x}{4 4 x ^{2} + 8} d x

Take the constant out:

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

= \frac{24}{4} \cdot \int \frac{x}{4 x ^{2} + 8} d x

Simplify

= 6 \cdot \int \frac{x}{4 x ^{2} + 8} d x

Apply u-substitution

: \int \frac{1}{8 u} d u

= 6 \cdot \int \frac{1}{8 u} d u

Take the constant out:

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

= 6 \cdot \frac{1}{8} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

= 6 \cdot \frac{1}{8} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = 4 x^{2} + 8

= 6 \cdot \frac{1}{8} \cdot 2 (4 x^{2} + 8)^{\frac{1}{2}}

Simplify

6 \cdot \frac{1}{8} \cdot 2 (4 x^{2} + 8)^{\frac{1}{2}} : 3 x^{2} + 2

= 3 x^{2} + 2

Add a constant to the solution

= 3 x^{2} + 2 + C

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What is the integral of (24x)/(4sqrt(4x^2+8)) ?
The integral of (24x)/(4sqrt(4x^2+8)) is 3sqrt(x^2+2)+C