What is the integral of sqrt(x^2+11) ?

The integral of sqrt(x^2+11) is 11(1/22 xsqrt(11+x^2)+1/2 ln| 1/(sqrt(11))x+sqrt(1/11 (11+x^2))|)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(x^2+11)

\int x^{2} + 11 d x

11 (\frac{1}{22} x 11 + x^{2} + \frac{1}{2} ln \frac{1}{11} x + \frac{1}{11} (11 + x^{2})) + C

Solution steps

\int x^{2} + 11 d x

Apply Trigonometric Substitution

: \int 11 sec^{3} (u) d u

= \int 11 sec^{3} (u) d u

Take the constant out:

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

= 11 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 11 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= 11 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{11} x)

= 11 \frac{sec ^{2} ( arctan ( \frac{1}{11} x ) ) sin ( arctan ( \frac{1}{11} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{11} x)) + sec (arctan (\frac{1}{11} x))

Simplify

11 \frac{sec ^{2} ( arctan ( \frac{1}{11} x ) ) sin ( arctan ( \frac{1}{11} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{11} x)) + sec (arctan (\frac{1}{11} x)) : 11 (\frac{1}{22} x 11 + x^{2} + \frac{1}{2} ln \frac{1}{11} x + \frac{1}{11} (11 + x^{2}))

= 11 (\frac{1}{22} x 11 + x^{2} + \frac{1}{2} ln \frac{1}{11} x + \frac{1}{11} (11 + x^{2}))

Add a constant to the solution

= 11 (\frac{1}{22} x 11 + x^{2} + \frac{1}{2} ln \frac{1}{11} x + \frac{1}{11} (11 + x^{2})) + C

\int \frac{80}{x ( 20 - x )} d x

\int \frac{1}{x x ^{2} + 25} d x

\int \frac{2 x - 7}{x ^{2} - 2 x - 8} d x

\int (1 - 3 x) 6 x^{5} d x

\int (\frac{1}{4} x^{3} + 3 x^{2} - \frac{3}{5} x + 4) d x

What is the integral of sqrt(x^2+11) ?
The integral of sqrt(x^2+11) is 11(1/22 xsqrt(11+x^2)+1/2 ln| 1/(sqrt(11))x+sqrt(1/11 (11+x^2))|)+C