What is the integral of 4/x+x/(x^2+5) ?

The integral of 4/x+x/(x^2+5) is 4ln|x|+1/2 ln|x^2+5|+C

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integral of 4/x+x/(x^2+5)

\int \frac{4}{x} + \frac{x}{x ^{2} + 5} d x

4 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 5 + C

Solution steps

\int \frac{4}{x} + \frac{x}{x ^{2} + 5} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{4}{x} d x + \int \frac{x}{x ^{2} + 5} d x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

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

= 4 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 5

Add a constant to the solution

= 4 ln ∣ x ∣ + \frac{1}{2} ln x^{2} + 5 + C

y = 3 - 9 x^{2}

\frac{d y}{d x}, e^{x} y = 8

\int sin (t) \cdot e^{- s t} d t

y^{^{'}} = \frac{( x ^{2} y + y ^{3} )}{x ^{3}}

\int_{1}^{5} \frac{ln ( x )}{x ^{3}} d x

What is the integral of 4/x+x/(x^2+5) ?
The integral of 4/x+x/(x^2+5) is 4ln|x|+1/2 ln|x^2+5|+C