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

The integral of ((sqrt(x)+6)^7)/(2sqrt(x)) is 1/8 (sqrt(x)+6)^8+C

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integral of ((sqrt(x)+6)^7)/(2sqrt(x))

\int \frac{( x + 6 ) ^{7}}{2 x} d x

\frac{1}{8} (x + 6)^{8} + C

Solution steps

\int \frac{( x + 6 ) ^{7}}{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 + 6 ) ^{7}}{x} d x

Apply u-substitution

: \int 2 u^{7} d u

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

Apply the Power Rule

: \frac{u ^{8}}{8}

= \frac{1}{2} \cdot 2 \cdot \frac{u ^{8}}{8}

Substitute back

u = x + 6

= \frac{1}{2} \cdot 2 \cdot \frac{( x + 6 ) ^{8}}{8}

Simplify

\frac{1}{2} \cdot 2 \cdot \frac{( x + 6 ) ^{8}}{8} : \frac{1}{8} (x + 6)^{8}

= \frac{1}{8} (x + 6)^{8}

Add a constant to the solution

= \frac{1}{8} (x + 6)^{8} + C

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What is the integral of ((sqrt(x)+6)^7)/(2sqrt(x)) ?
The integral of ((sqrt(x)+6)^7)/(2sqrt(x)) is 1/8 (sqrt(x)+6)^8+C