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

The integral of ((sqrt(x)+1)^3)/(sqrt(x)) is 1/2 (sqrt(x)+1)^4+C

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

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

\frac{1}{2} (x + 1)^{4} + C

Solution steps

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

Apply u-substitution

: \int 2 u^{3} d u

= \int 2 u^{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 2 \cdot \frac{u ^{4}}{4}

Substitute back

u = x + 1

= 2 \cdot \frac{( x + 1 ) ^{4}}{4}

Simplify

2 \cdot \frac{( x + 1 ) ^{4}}{4} : \frac{1}{2} (x + 1)^{4}

= \frac{1}{2} (x + 1)^{4}

Add a constant to the solution

= \frac{1}{2} (x + 1)^{4} + C

\frac{d y}{d x} = y, y (0) = 1

\frac{d y}{d x} = 8 x^{4} (6 + y^{2})^{\frac{3}{2}}

x \to 0 lim (\frac{e ^{x} - 1}{5 x})

y^{^{''}} - 2 y^{^{'}} + y = 3 e^{x}

\int \frac{9 x}{x ^{3} + 1} d x

What is the integral of ((sqrt(x)+1)^3)/(sqrt(x)) ?
The integral of ((sqrt(x)+1)^3)/(sqrt(x)) is 1/2 (sqrt(x)+1)^4+C