What is the integral of x^9sqrt(x^5+3) ?

The integral of x^9sqrt(x^5+3) is 2/5 (1/5 (x^5+3)^{5/2}-(x^5+3)^{3/2})+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of x^9sqrt(x^5+3)

\int x^{9} x^{5} + 3 d x

\frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}}) + C

Solution steps

\int x^{9} x^{5} + 3 d x

Apply u-substitution

: \int \frac{2 ( u ^{2} - 3 ) u ^{2}}{5} d u

= \int \frac{2 ( u ^{2} - 3 ) u ^{2}}{5} d u

Take the constant out:

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

= \frac{2}{5} \cdot \int (u^{2} - 3) u^{2} d u

Expand

(u^{2} - 3) u^{2} : u^{4} - 3 u^{2}

= \frac{2}{5} \cdot \int u^{4} - 3 u^{2} 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{2}{5} (\int u^{4} d u - \int 3 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 3 u^{2} d u = u^{3}

= \frac{2}{5} (\frac{u ^{5}}{5} - u^{3})

Substitute back

u = x^{5} + 3

= \frac{2}{5} (\frac{( x ^{5} + 3 ) ^{5}}{5} - (x^{5} + 3)^{3})

Simplify

\frac{2}{5} (\frac{( x ^{5} + 3 ) ^{5}}{5} - (x^{5} + 3)^{3}) : \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}})

= \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}})

Add a constant to the solution

= \frac{2}{5} (\frac{1}{5} (x^{5} + 3)^{\frac{5}{2}} - (x^{5} + 3)^{\frac{3}{2}}) + C

\frac{d}{d x} (ln ((5 x)^{2}))

y^{^{''}} + 7 y^{^{'}} = 0

\frac{d}{d x} (cot (3 - 2 x))

\frac{d ^{2} y}{d t ^{2}} + 5 \frac{d y}{d t} + 6 y = 1

x y^{2} y^{^{'}} = y^{3} - 4 x^{3}, y (1) = 4

What is the integral of x^9sqrt(x^5+3) ?
The integral of x^9sqrt(x^5+3) is 2/5 (1/5 (x^5+3)^{5/2}-(x^5+3)^{3/2})+C