What is the integral of (x^3+25)/(x^2+25) ?

The integral of (x^3+25)/(x^2+25) is 5(5/2 ((x^2)/(25)+1-ln|(x^2)/(25)+1|)+arctan(x/5))+C

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integral of (x^3+25)/(x^2+25)

\int \frac{x ^{3} + 25}{x ^{2} + 25} d x

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

Solution steps

\int \frac{x ^{3} + 25}{x ^{2} + 25} d x

Apply Integral Substitution

= \int \frac{5 ( 5 u ^{3} + 1 )}{u ^{2} + 1} d u

Take the constant out:

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

= 5 \cdot \int \frac{5 u ^{3} + 1}{u ^{2} + 1} d u

Expand

\frac{5 u ^{3} + 1}{u ^{2} + 1} : \frac{5 u ^{3}}{u ^{2} + 1} + \frac{1}{u ^{2} + 1}

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{5 u ^{3}}{u ^{2} + 1} d u + \int \frac{1}{u ^{2} + 1} d u)

\int \frac{5 u ^{3}}{u ^{2} + 1} d u = \frac{5}{2} (u^{2} + 1 - ln u^{2} + 1)

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

= 5 (\frac{5}{2} (u^{2} + 1 - ln u^{2} + 1) + arctan (u))

Substitute back

u = \frac{x}{5}

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

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

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

Add a constant to the solution

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

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

\frac{d}{d x} (\frac{2 x}{5})

y^{^{'}} = \frac{( x ^{2} - y ^{2} )}{x y}, y (2) = 5

\frac{d}{d x} ((\frac{3 x sin ( x )}{3 x + 1})^{3})

\frac{d}{d x} (y + x y)

What is the integral of (x^3+25)/(x^2+25) ?
The integral of (x^3+25)/(x^2+25) is 5(5/2 ((x^2)/(25)+1-ln|(x^2)/(25)+1|)+arctan(x/5))+C