What is the integral of (3x)/(sqrt(4x+7)) ?

The integral of (3x)/(sqrt(4x+7)) is 1/4 (2x-7)sqrt(4x+7)+C

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integral of (3x)/(sqrt(4x+7))

\int \frac{3 x}{4 x + 7} d x

\frac{1}{4} (2 x - 7) 4 x + 7 + C

Solution steps

\int \frac{3 x}{4 x + 7} d x

Take the constant out:

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

= 3 \cdot \int \frac{x}{4 x + 7} d x

Apply u-substitution

: \int \frac{u ^{2} - 7}{8} d u

= 3 \cdot \int \frac{u ^{2} - 7}{8} d u

Take the constant out:

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

= 3 \cdot \frac{1}{8} \cdot \int u^{2} - 7 d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 3 \cdot \frac{1}{8} (\int u^{2} d u - \int 7 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 7 d u = 7 u

= 3 \cdot \frac{1}{8} (\frac{u ^{3}}{3} - 7 u)

Substitute back

u = 4 x + 7

= 3 \cdot \frac{1}{8} (\frac{( 4 x + 7 ) ^{3}}{3} - 7 4 x + 7)

Simplify

3 \cdot \frac{1}{8} (\frac{( 4 x + 7 ) ^{3}}{3} - 7 4 x + 7) : \frac{1}{4} (2 x - 7) 4 x + 7

= \frac{1}{4} (2 x - 7) 4 x + 7

Add a constant to the solution

= \frac{1}{4} (2 x - 7) 4 x + 7 + C

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What is the integral of (3x)/(sqrt(4x+7)) ?
The integral of (3x)/(sqrt(4x+7)) is 1/4 (2x-7)sqrt(4x+7)+C