What is the integral of xsqrt(x^2+8x+21) ?

The integral of xsqrt(x^2+8x+21) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xsqrt(x^2+8x+21)

\int x x^{2} + 8 x + 21 d x

\frac{1}{3} (x^{2} + 8 x + 21)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) x^{2} + 8 x + 21 + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{x ^{2} + 8 x + 21}{5}) + C

Solution steps

\int x x^{2} + 8 x + 21 d x

Complete the square

x^{2} + 8 x + 21 : (x + 4)^{2} + 5

= \int x (x + 4)^{2} + 5 d x

Apply u-substitution

: \int (u - 4) u^{2} + 5 d u

= \int (u - 4) u^{2} + 5 d u

Expand

(u - 4) u^{2} + 5 : u u^{2} + 5 - 4 u^{2} + 5

= \int u u^{2} + 5 - 4 u^{2} + 5 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 u^{2} + 5 d u - \int 4 u^{2} + 5 d u

\int u u^{2} + 5 d u = \frac{1}{3} (u^{2} + 5)^{\frac{3}{2}}

\int 4 u^{2} + 5 d u = 20 (\frac{1}{10} u 5 + u^{2} + \frac{1}{2} ln \frac{1}{5} u + \frac{1}{5} (5 + u^{2}))

= \frac{1}{3} (u^{2} + 5)^{\frac{3}{2}} - 20 (\frac{1}{10} u 5 + u^{2} + \frac{1}{2} ln \frac{1}{5} u + \frac{1}{5} (5 + u^{2}))

Substitute back

u = x + 4

= \frac{1}{3} ((x + 4)^{2} + 5)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) 5 + (x + 4)^{2} + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{1}{5} (5 + (x + 4)^{2}))

Simplify

\frac{1}{3} ((x + 4)^{2} + 5)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) 5 + (x + 4)^{2} + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{1}{5} (5 + (x + 4)^{2})) : \frac{1}{3} (x^{2} + 8 x + 21)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) x^{2} + 8 x + 21 + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{x ^{2} + 8 x + 21}{5})

= \frac{1}{3} (x^{2} + 8 x + 21)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) x^{2} + 8 x + 21 + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{x ^{2} + 8 x + 21}{5})

Add a constant to the solution

= \frac{1}{3} (x^{2} + 8 x + 21)^{\frac{3}{2}} - 20 (\frac{1}{10} (x + 4) x^{2} + 8 x + 21 + \frac{1}{2} ln \frac{1}{5} (x + 4) + \frac{x ^{2} + 8 x + 21}{5}) + C

\int 324 - 9 x^{2} d x

\int 6 x - 5 d x

\int cosh (4 x - 1) d x

\int 2 y^{\frac{1}{2}} d y

\int (sec (2 x) - csc (\frac{x}{2})) d x