What is the integral of (sqrt(x^2+25))/(9x^2) ?

The integral of (sqrt(x^2+25))/(9x^2) is 1/9 (-(sqrt(x^2+25))/x+ln(1/5 |x+sqrt(25+x^2)|))+C

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integral of (sqrt(x^2+25))/(9x^2)

\int \frac{x ^{2} + 25}{9 x ^{2}} d x

\frac{1}{9} (- \frac{x ^{2} + 25}{x} + ln (\frac{1}{5} x + 25 + x^{2})) + C

Solution steps

\int \frac{x ^{2} + 25}{9 x ^{2}} d x

Take the constant out:

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

= \frac{1}{9} \cdot \int \frac{x ^{2} + 25}{x ^{2}} d x

Apply Integration By Parts

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

= \frac{1}{9} (- \frac{x ^{2} + 25}{x} - \int - \frac{1}{x ^{2} + 25} d x)

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

= \frac{1}{9} (- \frac{x ^{2} + 25}{x} - (- ln (\frac{1}{5} x + 25 + x^{2})))

Simplify

= \frac{1}{9} (- \frac{x ^{2} + 25}{x} + ln (\frac{1}{5} x + 25 + x^{2}))

Add a constant to the solution

= \frac{1}{9} (- \frac{x ^{2} + 25}{x} + ln (\frac{1}{5} x + 25 + x^{2})) + C

x \to 1 + lim (\frac{x + 5}{x + 2})

\frac{\partial}{\partial x} (ln (4 y))

y^{^{''}} - 5 y^{^{'}} + 4 y = - sin (5 t)

y^{^{'}} = \frac{x}{( 1 + 2 y )}

x, x - 2, [0, 4]

What is the integral of (sqrt(x^2+25))/(9x^2) ?
The integral of (sqrt(x^2+25))/(9x^2) is 1/9 (-(sqrt(x^2+25))/x+ln(1/5 |x+sqrt(25+x^2)|))+C