What is the integral of (9x)^5sqrt((9x)^3+1) ?

The integral of (9x)^5sqrt((9x)^3+1) is 1/27 (2/5 (729x^3+1)^{5/2}-2/3 (729x^3+1)^{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 (9x)^5sqrt((9x)^3+1)

\int (9 x)^{5} (9 x)^{3} + 1 d x

\frac{1}{27} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}}) + C

Solution steps

\int (9 x)^{5} (9 x)^{3} + 1 d x

Simplify

(9 x)^{5} (9 x)^{3} + 1 : 59049 x^{5} 729 x^{3} + 1

= \int 59049 x^{5} 729 x^{3} + 1 d x

Take the constant out:

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

= 59049 \cdot \int x^{5} 729 x^{3} + 1 d x

Apply u-substitution

: \int \frac{u ( u - 1 )}{1594323} d u

= 59049 \cdot \int \frac{u ( u - 1 )}{1594323} d u

Take the constant out:

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

= 59049 \cdot \frac{1}{1594323} \cdot \int u (u - 1) d u

Expand

u (u - 1) : u^{\frac{3}{2}} - u

= 59049 \cdot \frac{1}{1594323} \cdot \int u^{\frac{3}{2}} - u 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

= 59049 \cdot \frac{1}{1594323} (\int u^{\frac{3}{2}} d u - \int u d u)

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

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

= 59049 \cdot \frac{1}{1594323} (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{3} u^{\frac{3}{2}})

Substitute back

u = 729 x^{3} + 1

= 59049 \cdot \frac{1}{1594323} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}})

Simplify

59049 \cdot \frac{1}{1594323} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}}) : \frac{1}{27} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}})

= \frac{1}{27} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{27} (\frac{2}{5} (729 x^{3} + 1)^{\frac{5}{2}} - \frac{2}{3} (729 x^{3} + 1)^{\frac{3}{2}}) + C

\int \frac{x ^{2} - 3}{( x ^{2} + 1 ) ( x - 1 )} d x

t \frac{d y}{d t} = t^{3} + 18 t^{3} y

\int cos (x) (1 + 4 sin^{2} (x)) d x

\frac{d}{d x} (3 1 - x^{2})

f (x) = x, at x = 25

What is the integral of (9x)^5sqrt((9x)^3+1) ?
The integral of (9x)^5sqrt((9x)^3+1) is 1/27 (2/5 (729x^3+1)^{5/2}-2/3 (729x^3+1)^{3/2})+C