What is the integral of 7u^7sqrt(u^8+7) ?

The integral of 7u^7sqrt(u^8+7) is 7/12 (u^8+7)^{3/2}+C

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integral of 7u^7sqrt(u^8+7)

\int 7 u^{7} u^{8} + 7 d u

\frac{7}{12} (u^{8} + 7)^{\frac{3}{2}} + C

Solution steps

\int 7 u^{7} u^{8} + 7 d u

Take the constant out:

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

= 7 \cdot \int u^{7} u^{8} + 7 d u

Apply u-substitution

: \int \frac{v}{8} d v

= 7 \cdot \int \frac{v}{8} d v

Take the constant out:

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

= 7 \cdot \frac{1}{8} \cdot \int v d v

Apply the Power Rule

: \frac{2}{3} v^{\frac{3}{2}}

= 7 \cdot \frac{1}{8} \cdot \frac{2}{3} v^{\frac{3}{2}}

Substitute back

v = u^{8} + 7

= 7 \cdot \frac{1}{8} \cdot \frac{2}{3} (u^{8} + 7)^{\frac{3}{2}}

Simplify

7 \cdot \frac{1}{8} \cdot \frac{2}{3} (u^{8} + 7)^{\frac{3}{2}} : \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}}

= \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}}

Add a constant to the solution

= \frac{7}{12} (u^{8} + 7)^{\frac{3}{2}} + C

x \to 0 + lim (\frac{5}{x - x ^{2}})

x \to \frac{π}{2} lim (sin (\frac{1}{x}))

7 x^{2}, x^{2} + 1

\int e^{x} 3 + e^{x} d x

\frac{\partial}{\partial y} (sin (x) + x^{2} e^{y} - 1)

What is the integral of 7u^7sqrt(u^8+7) ?
The integral of 7u^7sqrt(u^8+7) is 7/12 (u^8+7)^{3/2}+C