What is the integral of ((sqrt(x)+1)^6)/(2sqrt(x)) ?

The integral of ((sqrt(x)+1)^6)/(2sqrt(x)) is 1/7 (sqrt(x)+1)^7+C

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integral of ((sqrt(x)+1)^6)/(2sqrt(x))

\int \frac{( x + 1 ) ^{6}}{2 x} d x

\frac{1}{7} (x + 1)^{7} + C

Solution steps

\int \frac{( x + 1 ) ^{6}}{2 x} d x

Take the constant out:

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

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

Apply u-substitution

: \int 2 u^{6} d u

= \frac{1}{2} \cdot \int 2 u^{6} d u

Take the constant out:

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

= \frac{1}{2} \cdot 2 \cdot \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= \frac{1}{2} \cdot 2 \cdot \frac{u ^{7}}{7}

Substitute back

u = x + 1

= \frac{1}{2} \cdot 2 \cdot \frac{( x + 1 ) ^{7}}{7}

Simplify

\frac{1}{2} \cdot 2 \cdot \frac{( x + 1 ) ^{7}}{7} : \frac{1}{7} (x + 1)^{7}

= \frac{1}{7} (x + 1)^{7}

Add a constant to the solution

= \frac{1}{7} (x + 1)^{7} + C

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What is the integral of ((sqrt(x)+1)^6)/(2sqrt(x)) ?
The integral of ((sqrt(x)+1)^6)/(2sqrt(x)) is 1/7 (sqrt(x)+1)^7+C