What is the integral of ((1+x)^3)/x ?

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

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integral of ((1+x)^3)/x

integral

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

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

Solution steps

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

Expand

\frac{( 1 + x ) ^{3}}{x} : \frac{1}{x} + 3 + 3 x + x^{2}

= \int \frac{1}{x} + 3 + 3 x + x^{2} 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} d x + \int 3 d x + \int 3 x d x + \int x^{2} d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int 3 d x = 3 x

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

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

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

Add a constant to the solution

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

(ln (x))^{4}

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

\frac{u}{( 1 + u ^{2} )}

\frac{2 tan ( x )}{( 1 + 2 tan ^{2} ( x ) )}

\frac{( x + 1 ) ^{2}}{( x - 1 ) ^{2}}

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