What is the integral of (1-2x)^5 ?

The integral of (1-2x)^5 is -1/12 (1-2x)^6+C

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integral of (1-2x)^5

\int (1 - 2 x)^{5} d x

- \frac{1}{12} (1 - 2 x)^{6} + C

Solution steps

\int (1 - 2 x)^{5} d x

Apply u-substitution

: \int - \frac{u ^{5}}{2} d u

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

Take the constant out:

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

= - \frac{1}{2} \cdot \int u^{5} d u

Apply the Power Rule

: \frac{u ^{6}}{6}

= - \frac{1}{2} \cdot \frac{u ^{6}}{6}

Substitute back

u = 1 - 2 x

= - \frac{1}{2} \cdot \frac{( 1 - 2 x ) ^{6}}{6}

Simplify

- \frac{1}{2} \cdot \frac{( 1 - 2 x ) ^{6}}{6} : - \frac{1}{12} (1 - 2 x)^{6}

= - \frac{1}{12} (1 - 2 x)^{6}

Add a constant to the solution

= - \frac{1}{12} (1 - 2 x)^{6} + C

y = e^{x}, y = 7 e^{- x}, x = 0

\int ln (x + x^{2} - 1) d x

y^{^{'}} = - 100 y + 101 cos (t) + 99 sin (t), y (0) = 1

\int (\frac{1}{x} - \frac{2}{9 x ^{2} + 1}) d x

\frac{d}{d x} ((5 x + 1)^{3})