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

The integral of (x^7+1)/(sqrt(x^8+8x)) is 1/4 sqrt(x^8+8x)+C

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integral of (x^7+1)/(sqrt(x^8+8x))

\int \frac{x ^{7} + 1}{x ^{8} + 8 x} d x

\frac{1}{4} x^{8} + 8 x + C

Solution steps

\int \frac{x ^{7} + 1}{x ^{8} + 8 x} d x

Apply u-substitution

= \int \frac{1}{8 u} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

Substitute back

u = x^{8} + 8 x

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

Simplify

\frac{1}{8} \cdot 2 (x^{8} + 8 x)^{\frac{1}{2}} : \frac{1}{4} x^{8} + 8 x

= \frac{1}{4} x^{8} + 8 x

Add a constant to the solution

= \frac{1}{4} x^{8} + 8 x + C

What is the integral of (x^7+1)/(sqrt(x^8+8x)) ?
The integral of (x^7+1)/(sqrt(x^8+8x)) is 1/4 sqrt(x^8+8x)+C