What is the integral of (a-x)^3 ?

The integral of (a-x)^3 is -((a-x)^4)/4+C

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integral of (a-x)^3

\int (a - x)^{3} d x

- \frac{( a - x ) ^{4}}{4} + C

Solution steps

\int (a - x)^{3} d x

Apply u-substitution

: \int - u^{3} d u

= \int - u^{3} d u

Take the constant out:

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

= - \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= - \frac{u ^{4}}{4}

Substitute back

u = a - x

= - \frac{( a - x ) ^{4}}{4}

Add a constant to the solution

= - \frac{( a - x ) ^{4}}{4} + C

\int (3 + 0.5 t) d t

\frac{\partial}{\partial x} (ln (2 x) + 5 y^{3})

\frac{\partial}{\partial y} (ln (x + \frac{y}{2 x}))

\int_{0}^{1} \frac{1}{x ^{\frac{1}{2}}} d x

y^{^{'}} = - 10 y