What is the integral of (10x+2)sqrt(5x^2+2x) ?

The integral of (10x+2)sqrt(5x^2+2x) is 2/(15sqrt(5))(25x^2+10x)^{3/2}+C

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integral of (10x+2)sqrt(5x^2+2x)

\int (10 x + 2) 5 x^{2} + 2 x d x

\frac{2}{15 5} (25 x^{2} + 10 x)^{\frac{3}{2}} + C

Solution steps

\int (10 x + 2) 5 x^{2} + 2 x d x

(10 x + 2) 5 x^{2} + 2 x = 2 (5 x + 1) 5 x^{2} + 2 x

= \int 2 (5 x + 1) 5 x^{2} + 2 x d x

Take the constant out:

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

= 2 \cdot \int (5 x + 1) 5 x^{2} + 2 x d x

Apply u-substitution

: \int \frac{u u ^{2} - 1}{5 5} d u

= 2 \cdot \int \frac{u u ^{2} - 1}{5 5} d u

Take the constant out:

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

= 2 \cdot \frac{1}{5 5} \cdot \int u u^{2} - 1 d u

Apply u-substitution

: \int v^{2} d v

= 2 \cdot \frac{1}{5 5} \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= 2 \cdot \frac{1}{5 5} \cdot \frac{v ^{3}}{3}

Substitute back

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

Simplify

2 \cdot \frac{1}{5 5} \cdot \frac{( ( 5 x + 1 ) ^{2} - 1 ) ^{3}}{3} : \frac{2}{15 5} (25 x^{2} + 10 x)^{\frac{3}{2}}

= \frac{2}{15 5} (25 x^{2} + 10 x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{15 5} (25 x^{2} + 10 x)^{\frac{3}{2}} + C

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What is the integral of (10x+2)sqrt(5x^2+2x) ?
The integral of (10x+2)sqrt(5x^2+2x) is 2/(15sqrt(5))(25x^2+10x)^{3/2}+C