What is the integral of zsqrt(1-z) ?

The integral of zsqrt(1-z) is 2/5 (1-z)^{5/2}-2/3 (1-z)^{3/2}+C

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integral of zsqrt(1-z)

\int z 1 - z d z

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

Solution steps

\int z 1 - z d z

Apply u-substitution

: \int - (- u + 1) u d u

= \int - (- u + 1) u d u

Take the constant out:

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

= - \int (- u + 1) u d u

Expand

(- u + 1) u : - u^{\frac{3}{2}} + u

= - \int - u^{\frac{3}{2}} + u 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

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

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

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

= - (- \frac{2}{5} u^{\frac{5}{2}} + \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 1 - z

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

Simplify

= \frac{2}{5} (1 - z)^{\frac{5}{2}} - \frac{2}{3} (1 - z)^{\frac{3}{2}}

Add a constant to the solution

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

x \to 0 lim (\frac{x - 2}{∣ x - 2 ∣})

\int_{0}^{4} y - y + 2 d y

\frac{d}{d x} (4 x (x^{2} - 4))

\int \frac{1}{2 x ^{2} - x} d x

f (x) = 5 x - 7

What is the integral of zsqrt(1-z) ?
The integral of zsqrt(1-z) is 2/5 (1-z)^{5/2}-2/3 (1-z)^{3/2}+C