What is the integral of x(2x+7)^8 ?

The integral of x(2x+7)^8 is 1/360 (9(2x+7)^{10}-70(2x+7)^9)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x(2x+7)^8

\int x (2 x + 7)^{8} d x

\frac{1}{360} (9 (2 x + 7)^{10} - 70 (2 x + 7)^{9}) + C

Solution steps

\int x (2 x + 7)^{8} d x

Apply u-substitution

: \int \frac{u ^{8} ( u - 7 )}{4} d u

= \int \frac{u ^{8} ( u - 7 )}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int u^{8} (u - 7) d u

Expand

u^{8} (u - 7) : u^{9} - 7 u^{8}

= \frac{1}{4} \cdot \int u^{9} - 7 u^{8} 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

= \frac{1}{4} (\int u^{9} d u - \int 7 u^{8} d u)

\int u^{9} d u = \frac{u ^{10}}{10}

\int 7 u^{8} d u = \frac{7 u ^{9}}{9}

= \frac{1}{4} (\frac{u ^{10}}{10} - \frac{7 u ^{9}}{9})

Substitute back

u = 2 x + 7

= \frac{1}{4} (\frac{( 2 x + 7 ) ^{10}}{10} - \frac{7 ( 2 x + 7 ) ^{9}}{9})

Simplify

\frac{1}{4} (\frac{( 2 x + 7 ) ^{10}}{10} - \frac{7 ( 2 x + 7 ) ^{9}}{9}) : \frac{1}{360} (9 (2 x + 7)^{10} - 70 (2 x + 7)^{9})

= \frac{1}{360} (9 (2 x + 7)^{10} - 70 (2 x + 7)^{9})

Add a constant to the solution

= \frac{1}{360} (9 (2 x + 7)^{10} - 70 (2 x + 7)^{9}) + C

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

\int \frac{1}{6 x - 1} d x

\int x^{2} e^{b x} d x

sin (t + π)

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

What is the integral of x(2x+7)^8 ?
The integral of x(2x+7)^8 is 1/360 (9(2x+7)^{10}-70(2x+7)^9)+C