What is the integral of 5/4 sqrt(5z+7) ?

The integral of 5/4 sqrt(5z+7) is 1/6 (5z+7)^{3/2}+C

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integral of 5/4 sqrt(5z+7)

\int \frac{5}{4} 5 z + 7 d z

\frac{1}{6} (5 z + 7)^{\frac{3}{2}} + C

Solution steps

\int \frac{5}{4} 5 z + 7 d z

Take the constant out:

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

= \frac{5}{4} \cdot \int 5 z + 7 d z

Apply u-substitution

: \int \frac{u}{5} d u

= \frac{5}{4} \cdot \int \frac{u}{5} d u

Take the constant out:

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

= \frac{5}{4} \cdot \frac{1}{5} \cdot \int u d u

Apply the Power Rule

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

= \frac{5}{4} \cdot \frac{1}{5} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 5 z + 7

= \frac{5}{4} \cdot \frac{1}{5} \cdot \frac{2}{3} (5 z + 7)^{\frac{3}{2}}

Simplify

\frac{5}{4} \cdot \frac{1}{5} \cdot \frac{2}{3} (5 z + 7)^{\frac{3}{2}} : \frac{1}{6} (5 z + 7)^{\frac{3}{2}}

= \frac{1}{6} (5 z + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{6} (5 z + 7)^{\frac{3}{2}} + C

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What is the integral of 5/4 sqrt(5z+7) ?
The integral of 5/4 sqrt(5z+7) is 1/6 (5z+7)^{3/2}+C