What is the integral of (a+bx)^n ?

The integral of (a+bx)^n is ((a+bx)^{n+1})/(b(n+1))+C

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integral of (a+bx)^n

\int (a + b x)^{n} d x

\frac{( a + b x ) ^{n + 1}}{b ( n + 1 )} + C

Solution steps

\int (a + b x)^{n} d x

Apply u-substitution

: \int \frac{u ^{n}}{b} d u

= \int \frac{u ^{n}}{b} d u

Take the constant out:

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

= \frac{1}{b} \cdot \int u^{n} d u

Apply the Power Rule

: \frac{u ^{n + 1}}{n + 1}

= \frac{1}{b} \cdot \frac{u ^{n + 1}}{n + 1}

Substitute back

u = a + b x

= \frac{1}{b} \cdot \frac{( a + b x ) ^{n + 1}}{n + 1}

Simplify

\frac{1}{b} \cdot \frac{( a + b x ) ^{n + 1}}{n + 1} : \frac{( a + b x ) ^{n + 1}}{b ( n + 1 )}

= \frac{( a + b x ) ^{n + 1}}{b ( n + 1 )}

Add a constant to the solution

= \frac{( a + b x ) ^{n + 1}}{b ( n + 1 )} + C

\int (sin (x)) d x

\int arctan (2 y) d y

y = ln (\frac{1 + e ^{x}}{1 - e ^{x}})

\frac{d y}{d x} = - \frac{y ^{2}}{1 + y ^{2}} e^{(- x)}

n = 1 \sum \infty \frac{2 n}{7 n - 2}

What is the integral of (a+bx)^n ?
The integral of (a+bx)^n is ((a+bx)^{n+1})/(b(n+1))+C