What is the integral of x^7sqrt(ax^8-b) ?

The integral of x^7sqrt(ax^8-b) is ((ax^8-b)^{3/2})/(12a)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^7sqrt(ax^8-b)

\int x^{7} a x^{8} - b d x

\frac{( a x ^{8} - b ) ^{\frac{3}{2}}}{12 a} + C

Solution steps

\int x^{7} a x^{8} - b d x

Apply u-substitution

: \int \frac{u}{8 a} d u

= \int \frac{u}{8 a} d u

Take the constant out:

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

= \frac{1}{8 a} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{8 a} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = a x^{8} - b

= \frac{1}{8 a} \cdot \frac{2}{3} (a x^{8} - b)^{\frac{3}{2}}

Simplify

\frac{1}{8 a} \cdot \frac{2}{3} (a x^{8} - b)^{\frac{3}{2}} : \frac{( a x ^{8} - b ) ^{\frac{3}{2}}}{12 a}

= \frac{( a x ^{8} - b ) ^{\frac{3}{2}}}{12 a}

Add a constant to the solution

= \frac{( a x ^{8} - b ) ^{\frac{3}{2}}}{12 a} + C

\frac{d}{d x} (\frac{2 x}{x ^{2} + 2})

\frac{d}{d x} (tan^{2} (x))

x \to 0 lim (e^{4 x + 2})

f (x) = (1 + tan (2 x))^{3}

sin (- 5 x)

What is the integral of x^7sqrt(ax^8-b) ?
The integral of x^7sqrt(ax^8-b) is ((ax^8-b)^{3/2})/(12a)+C