What is the integral of ((x^3))/(sqrt(x^2+100)) ?

The integral of ((x^3))/(sqrt(x^2+100)) is 1/3 (x^2+100)^{3/2}-100sqrt(x^2+100)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((x^3))/(sqrt(x^2+100))

\int \frac{( x ^{3} )}{x ^{2} + 100} d x

\frac{1}{3} (x^{2} + 100)^{\frac{3}{2}} - 100 x^{2} + 100 + C

Solution steps

\int \frac{x ^{3}}{x ^{2} + 100} d x

Apply u-substitution

: \int u^{2} - 100 d u

= \int u^{2} - 100 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

= \int u^{2} d u - \int 100 d u

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

\int 100 d u = 100 u

= \frac{u ^{3}}{3} - 100 u

Substitute back

u = x^{2} + 100

= \frac{( x ^{2} + 100 ) ^{3}}{3} - 100 x^{2} + 100

Simplify

\frac{( x ^{2} + 100 ) ^{3}}{3} - 100 x^{2} + 100 : \frac{1}{3} (x^{2} + 100)^{\frac{3}{2}} - 100 x^{2} + 100

= \frac{1}{3} (x^{2} + 100)^{\frac{3}{2}} - 100 x^{2} + 100

Add a constant to the solution

= \frac{1}{3} (x^{2} + 100)^{\frac{3}{2}} - 100 x^{2} + 100 + C

e^{r x}

\frac{d}{d x} (- e^{- x} x^{2} + 6 e^{- x} x - 6 e^{- x})

\int cot^{3} (10 x) d x

2 \frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} + y = 1

\frac{d}{d x} (cos (x^{3} + 2 x))

What is the integral of ((x^3))/(sqrt(x^2+100)) ?
The integral of ((x^3))/(sqrt(x^2+100)) is 1/3 (x^2+100)^{3/2}-100sqrt(x^2+100)+C