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

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

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integral of (x^6+1)/(sqrt(x^7+7x))

\int \frac{x ^{6} + 1}{x ^{7} + 7 x} d x

\frac{2}{7} x^{7} + 7 x + C

Solution steps

\int \frac{x ^{6} + 1}{x ^{7} + 7 x} d x

Apply u-substitution

= \int \frac{1}{7 u} d u

Take the constant out:

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

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

Apply the Power Rule

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

Substitute back

u = x^{7} + 7 x

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

Simplify

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

= \frac{2}{7} x^{7} + 7 x

Add a constant to the solution

= \frac{2}{7} x^{7} + 7 x + C

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