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

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

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

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

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

Solution steps

\int \frac{( 1 + x ) ^{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 = 1 + x

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

Simplify

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

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

Add a constant to the solution

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

y^{^{'}} = tan (t) y - 5 t, y (0) = 3

\frac{d y}{d x} = \frac{e ^{7 x}}{4 y ^{3}}

\frac{\partial}{\partial x} (x^{\frac{1}{3}} \cdot y^{\frac{2}{3}})

y = (ln (x))^{4}

\int \frac{x ^{3}}{( 49 - 4 x ^{2} ) ^{\frac{3}{2}}} d x

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