What is the integral of (4x+2)sqrt(x^2+x) ?

The integral of (4x+2)sqrt(x^2+x) is 4/3 (x^2+x)^{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 (4x+2)sqrt(x^2+x)

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

\frac{4}{3} (x^{2} + x)^{\frac{3}{2}} + C

Solution steps

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

(4 x + 2) x^{2} + x = 2 (2 x + 1) x^{2} + x

= \int 2 (2 x + 1) x^{2} + x d x

Take the constant out:

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

= 2 \cdot \int (2 x + 1) x^{2} + x d x

Apply u-substitution

: \int u d u

= 2 \cdot \int u d u

Apply the Power Rule

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

= 2 \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{2} + x

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

Simplify

2 \cdot \frac{2}{3} (x^{2} + x)^{\frac{3}{2}} : \frac{4}{3} (x^{2} + x)^{\frac{3}{2}}

= \frac{4}{3} (x^{2} + x)^{\frac{3}{2}}

Add a constant to the solution

= \frac{4}{3} (x^{2} + x)^{\frac{3}{2}} + C

\int e^{i 2 x} d x

\int - 2 x 25 - x^{2} d x

\int \frac{1}{x ( 2 + x ^{8} )} d x

\int (csc^{2} (x) + 2) d x

\int \frac{3 x - 2}{x} d x

What is the integral of (4x+2)sqrt(x^2+x) ?
The integral of (4x+2)sqrt(x^2+x) is 4/3 (x^2+x)^{3/2}+C