What is the integral of 5/(sqrt(x^2+25)) ?

The integral of 5/(sqrt(x^2+25)) is 5ln(1/5 |x+sqrt(25+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 5/(sqrt(x^2+25))

\int \frac{5}{x ^{2} + 25} d x

5 ln (\frac{1}{5} x + 25 + x^{2}) + C

Solution steps

\int \frac{5}{x ^{2} + 25} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ^{2} + 25} d x

Apply Trigonometric Substitution

: \int sec (u) d u

= 5 \cdot \int sec (u) d u

Use the common integral:

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

= 5 ln ∣ tan (u) + sec (u) ∣

Substitute back

u = arctan (\frac{1}{5} x)

= 5 ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x))

Simplify

5 ln tan (arctan (\frac{1}{5} x)) + sec (arctan (\frac{1}{5} x)) : 5 ln (\frac{1}{5} x + 25 + x^{2})

= 5 ln (\frac{1}{5} x + 25 + x^{2})

Add a constant to the solution

= 5 ln (\frac{1}{5} x + 25 + x^{2}) + C

\int \frac{2 + sin ( x )}{x} d x

\int e^{\frac{- x}{100}} d x

\int sec^{2} (1 - 2 x) tan^{2} (1 - 2 x) d x

\int sin (ln (x^{3})) d x

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

What is the integral of 5/(sqrt(x^2+25)) ?
The integral of 5/(sqrt(x^2+25)) is 5ln(1/5 |x+sqrt(25+x^2)|)+C