What is the integral of (4x^4+x+1)/(x^5+x^4) ?

The integral of (4x^4+x+1)/(x^5+x^4) is -1/(3x^3)+4ln|x+1|+C

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integral of (4x^4+x+1)/(x^5+x^4)

\int \frac{4 x ^{4} + x + 1}{x ^{5} + x ^{4}} d x

- \frac{1}{3 x ^{3}} + 4 ln ∣ x + 1 ∣ + C

Solution steps

\int \frac{4 x ^{4} + x + 1}{x ^{5} + x ^{4}} d x

Take the partial fraction of

\frac{4 x ^{4} + x + 1}{x ^{5} + x ^{4}} : \frac{1}{x ^{4}} + \frac{4}{x + 1}

= \int \frac{1}{x ^{4}} + \frac{4}{x + 1} 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{1}{x ^{4}} d x + \int \frac{4}{x + 1} d x

\int \frac{1}{x ^{4}} d x = - \frac{1}{3 x ^{3}}

\int \frac{4}{x + 1} d x = 4 ln ∣ x + 1 ∣

= - \frac{1}{3 x ^{3}} + 4 ln ∣ x + 1 ∣

Add a constant to the solution

= - \frac{1}{3 x ^{3}} + 4 ln ∣ x + 1 ∣ + C

ln (225 sin^{2} (x))

y = \frac{1}{x}

\int_{1}^{\infty} x^{- 3} d x

y^{^{'}} - 5 y = 4 e^{7 t}

x \to 0 lim (x^{2} + a x + 2)

What is the integral of (4x^4+x+1)/(x^5+x^4) ?
The integral of (4x^4+x+1)/(x^5+x^4) is -1/(3x^3)+4ln|x+1|+C