What is the integral of (x^3+4x^2-5)/(x^3) ?

The integral of (x^3+4x^2-5)/(x^3) is x+4ln|x|+5/(2x^2)+C

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integral of (x^3+4x^2-5)/(x^3)

\int \frac{x ^{3} + 4 x ^{2} - 5}{x ^{3}} d x

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

Solution steps

\int \frac{x ^{3} + 4 x ^{2} - 5}{x ^{3}} d x

Expand

\frac{x ^{3} + 4 x ^{2} - 5}{x ^{3}} : 1 + \frac{4}{x} - \frac{5}{x ^{3}}

= \int 1 + \frac{4}{x} - \frac{5}{x ^{3}} 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 1 d x + \int \frac{4}{x} d x - \int \frac{5}{x ^{3}} d x

\int 1 d x = x

\int \frac{4}{x} d x = 4 ln ∣ x ∣

\int \frac{5}{x ^{3}} d x = - \frac{5}{2 x ^{2}}

= x + 4 ln ∣ x ∣ - (- \frac{5}{2 x ^{2}})

Simplify

= x + 4 ln ∣ x ∣ + \frac{5}{2 x ^{2}}

Add a constant to the solution

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

\frac{d}{d x} (a cos (x) + x^{2} sin (x))

3 x y - 2 x + 16 = 0, (2, - 2)

x \to 0 lim (\frac{sin ( \frac{3}{5} ) x}{x})

\int cos^{4} (5 x) d x

\frac{d}{d x} (sin (2 x) + 10)

What is the integral of (x^3+4x^2-5)/(x^3) ?
The integral of (x^3+4x^2-5)/(x^3) is x+4ln|x|+5/(2x^2)+C