What is the integral of 250sqrt((x+4)^3) ?

The integral of 250sqrt((x+4)^3) is 100(x+4)^{5/2}+C

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integral of 250sqrt((x+4)^3)

\int 250 (x + 4)^{3} d x

100 (x + 4)^{\frac{5}{2}} + C

Solution steps

\int 250 (x + 4)^{3} d x

Take the constant out:

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

= 250 \cdot \int (x + 4)^{3} d x

(x + 4)^{3} = (x + 4)^{\frac{3}{2}},

assuming

(x + 4) \geq 0

= 250 \cdot \int (x + 4)^{\frac{3}{2}} d x

Apply u-substitution

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

= 250 \cdot \int u^{\frac{3}{2}} d u

Apply the Power Rule

: \frac{2}{5} u^{\frac{5}{2}}

= 250 \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = x + 4

= 250 \cdot \frac{2}{5} (x + 4)^{\frac{5}{2}}

Simplify

250 \cdot \frac{2}{5} (x + 4)^{\frac{5}{2}} : 100 (x + 4)^{\frac{5}{2}}

= 100 (x + 4)^{\frac{5}{2}}

Add a constant to the solution

= 100 (x + 4)^{\frac{5}{2}} + C

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What is the integral of 250sqrt((x+4)^3) ?
The integral of 250sqrt((x+4)^3) is 100(x+4)^{5/2}+C