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

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

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

integral

h (x) = \frac{4 x ^{3} + 2 x + 5}{x}

\frac{4 x ^{3}}{3} + 2 x + 5 ln ∣ x ∣ + C

Solution steps

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

Expand

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

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

\int 4 x^{2} d x = \frac{4 x ^{3}}{3}

\int 2 d x = 2 x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= \frac{4 x ^{3}}{3} + 2 x + 5 ln ∣ x ∣

Add a constant to the solution

= \frac{4 x ^{3}}{3} + 2 x + 5 ln ∣ x ∣ + C

\frac{t ^{2}}{2} + \frac{t ^{3}}{3} + \frac{t ^{4}}{4}

(\frac{sin ( x )}{x})^{2}

sinh (5 x + 2)

\frac{- 4}{( x ^{2} + 4 ) ^{2}}

\frac{3 x ^{2} + 4}{( x ^{3} + 4 x ) ^{4}}

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