What is the integral of sqrt(x^2+15) ?

The integral of sqrt(x^2+15) is 15(1/30 xsqrt(15+x^2)+1/2 ln| 1/(sqrt(15))x+sqrt(1/15 (15+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+15)

\int x^{2} + 15 d x

15 (\frac{1}{30} x 15 + x^{2} + \frac{1}{2} ln \frac{1}{15} x + \frac{1}{15} (15 + x^{2})) + C

Solution steps

\int x^{2} + 15 d x

Apply Trigonometric Substitution

: \int 15 sec^{3} (u) d u

= \int 15 sec^{3} (u) d u

Take the constant out:

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

= 15 \cdot \int sec^{3} (u) d u

Apply Integral Reduction

= 15 (\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) ∣

= 15 (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = arctan (\frac{1}{15} x)

= 15 \frac{sec ^{2} ( arctan ( \frac{1}{15} x ) ) sin ( arctan ( \frac{1}{15} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{15} x)) + sec (arctan (\frac{1}{15} x))

Simplify

15 \frac{sec ^{2} ( arctan ( \frac{1}{15} x ) ) sin ( arctan ( \frac{1}{15} x ) )}{2} + \frac{1}{2} ln tan (arctan (\frac{1}{15} x)) + sec (arctan (\frac{1}{15} x)) : 15 (\frac{1}{30} x 15 + x^{2} + \frac{1}{2} ln \frac{1}{15} x + \frac{1}{15} (15 + x^{2}))

= 15 (\frac{1}{30} x 15 + x^{2} + \frac{1}{2} ln \frac{1}{15} x + \frac{1}{15} (15 + x^{2}))

Add a constant to the solution

= 15 (\frac{1}{30} x 15 + x^{2} + \frac{1}{2} ln \frac{1}{15} x + \frac{1}{15} (15 + x^{2})) + C

9 - x^{2}

\int \frac{2}{3 x ^{4}} d x

\int tan (5 x + 1) d x

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

\int \frac{x ^{3}}{81 - x ^{4}} d x

What is the integral of sqrt(x^2+15) ?
The integral of sqrt(x^2+15) is 15(1/30 xsqrt(15+x^2)+1/2 ln| 1/(sqrt(15))x+sqrt(1/15 (15+x^2))|)+C