What is the integral of ((sqrt(x)+1)^5)/(sqrt(x)) ?

The integral of ((sqrt(x)+1)^5)/(sqrt(x)) is 1/3 (sqrt(x)+1)^6+C

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integral of ((sqrt(x)+1)^5)/(sqrt(x))

\int \frac{( x + 1 ) ^{5}}{x} d x

\frac{1}{3} (x + 1)^{6} + C

Solution steps

\int \frac{( x + 1 ) ^{5}}{x} d x

Apply u-substitution

= \int 2 u^{5} d u

Take the constant out:

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

= 2 \cdot \int u^{5} d u

Apply the Power Rule

= 2 \cdot \frac{u ^{6}}{6}

Substitute back

u = x + 1

= 2 \cdot \frac{( x + 1 ) ^{6}}{6}

Simplify

2 \cdot \frac{( x + 1 ) ^{6}}{6} : \frac{1}{3} (x + 1)^{6}

= \frac{1}{3} (x + 1)^{6}

Add a constant to the solution

= \frac{1}{3} (x + 1)^{6} + C

What is the integral of ((sqrt(x)+1)^5)/(sqrt(x)) ?
The integral of ((sqrt(x)+1)^5)/(sqrt(x)) is 1/3 (sqrt(x)+1)^6+C