What is the integral of sqrt(1-f(x)) ?

The integral of sqrt(1-f(x)) is -(2(1-xf)^{3/2})/(3x)+C

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integral of sqrt(1-f(x))

integral

1 - f (x)

- \frac{2 ( 1 - x f ) ^{\frac{3}{2}}}{3 x} + C

Solution steps

\int 1 - f x df

Apply u-substitution

: \int - \frac{u}{x} d u

= \int - \frac{u}{x} d u

Take the constant out:

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

= - \frac{1}{x} \cdot \int u d u

Apply the Power Rule

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

= - \frac{1}{x} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = 1 - x f

= - \frac{1}{x} \cdot \frac{2}{3} (1 - x f)^{\frac{3}{2}}

Simplify

- \frac{1}{x} \cdot \frac{2}{3} (1 - x f)^{\frac{3}{2}} : - \frac{2 ( 1 - x f ) ^{\frac{3}{2}}}{3 x}

= - \frac{2 ( 1 - x f ) ^{\frac{3}{2}}}{3 x}

Add a constant to the solution

= - \frac{2 ( 1 - x f ) ^{\frac{3}{2}}}{3 x} + C

3 x^{2} + 4 x

cos^{2} (3 x)

\frac{1 + e ^{- x}}{e ^{x}}

\frac{x ^{2}}{e ^{x}}

(x + 2)^{5}

What is the integral of sqrt(1-f(x)) ?
The integral of sqrt(1-f(x)) is -(2(1-xf)^{3/2})/(3x)+C