What is the integral of (3z)/(sqrt(3-7z)) ?

The integral of (3z)/(sqrt(3-7z)) is 6/49 (1/3 (3-7z)^{3/2}-3sqrt(3-7z))+C

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integral of (3z)/(sqrt(3-7z))

\int \frac{3 z}{3 - 7 z} d z

\frac{6}{49} (\frac{1}{3} (3 - 7 z)^{\frac{3}{2}} - 3 3 - 7 z) + C

Solution steps

\int \frac{3 z}{3 - 7 z} d z

Take the constant out:

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

= 3 \cdot \int \frac{z}{3 - 7 z} d z

Apply u-substitution

: \int \frac{2 ( u ^{2} - 3 )}{49} d u

= 3 \cdot \int \frac{2 ( u ^{2} - 3 )}{49} d u

Take the constant out:

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

= 3 \cdot \frac{2}{49} \cdot \int u^{2} - 3 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{2}{49} (\int u^{2} d u - \int 3 d u)

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

\int 3 d u = 3 u

= 3 \cdot \frac{2}{49} (\frac{u ^{3}}{3} - 3 u)

Substitute back

u = 3 - 7 z

= 3 \cdot \frac{2}{49} (\frac{( 3 - 7 z ) ^{3}}{3} - 3 3 - 7 z)

Simplify

3 \cdot \frac{2}{49} (\frac{( 3 - 7 z ) ^{3}}{3} - 3 3 - 7 z) : \frac{6}{49} (\frac{1}{3} (3 - 7 z)^{\frac{3}{2}} - 3 3 - 7 z)

= \frac{6}{49} (\frac{1}{3} (3 - 7 z)^{\frac{3}{2}} - 3 3 - 7 z)

Add a constant to the solution

= \frac{6}{49} (\frac{1}{3} (3 - 7 z)^{\frac{3}{2}} - 3 3 - 7 z) + C

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What is the integral of (3z)/(sqrt(3-7z)) ?
The integral of (3z)/(sqrt(3-7z)) is 6/49 (1/3 (3-7z)^{3/2}-3sqrt(3-7z))+C