What is the integral of (5sqrt(x^2-1))/x ?

The integral of (5sqrt(x^2-1))/x is 5(-arctan(sqrt(x^2-1))+sqrt(x^2-1))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (5sqrt(x^2-1))/x

\int \frac{5 x ^{2} - 1}{x} d x

5 (- arctan (x^{2} - 1) + x^{2} - 1) + C

Solution steps

\int \frac{5 x ^{2} - 1}{x} d x

Take the constant out:

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

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

Apply u-substitution

: \int \frac{u ^{2}}{u ^{2} + 1} d u

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

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= 5 \cdot \int - \frac{1}{u ^{2} + 1} + 1 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

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

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

\int 1 d u = u

= 5 (- arctan (u) + u)

Substitute back

u = x^{2} - 1

= 5 (- arctan (x^{2} - 1) + x^{2} - 1)

Add a constant to the solution

= 5 (- arctan (x^{2} - 1) + x^{2} - 1) + C

\int 9.8 x + 10 d x

\int (1 + u + u^{2}) d u

\int \frac{4 x - 7}{( x - 2 ) ^{2}} d x

\int 6 x^{5} e^{x} d x

\int (\frac{p}{5} - \frac{1}{2}) d p

What is the integral of (5sqrt(x^2-1))/x ?
The integral of (5sqrt(x^2-1))/x is 5(-arctan(sqrt(x^2-1))+sqrt(x^2-1))+C