What is the integral of (2ax+b)(ax^2+bx)^7 ?

The integral of (2ax+b)(ax^2+bx)^7 is ((ax^2+bx)^8)/8+C

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integral of (2ax+b)(ax^2+bx)^7

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

\frac{( a x ^{2} + b x ) ^{8}}{8} + C

Solution steps

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

Apply u-substitution

: \int u^{7} d u

= \int u^{7} d u

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{u ^{8}}{8}

Substitute back

u = a x^{2} + b x

= \frac{( a x ^{2} + b x ) ^{8}}{8}

Add a constant to the solution

= \frac{( a x ^{2} + b x ) ^{8}}{8} + C

\int e^{- 3 x} sin (2 x) d x

x^{2} 5^{x}

\frac{\partial}{\partial x} (ln (8 x^{2} + y^{2} + 3))

\frac{\partial}{\partial y} (15 x^{2} - 9 x y + 25 y^{2})

p (t) = 12 t e^{- \frac{t}{6}}

What is the integral of (2ax+b)(ax^2+bx)^7 ?
The integral of (2ax+b)(ax^2+bx)^7 is ((ax^2+bx)^8)/8+C