What is the integral of (x^6)/(sqrt(x^{14)-7)} ?

The integral of (x^6)/(sqrt(x^{14)-7)} is 1/7 ln|sqrt(1/7 (x^{14)-7)}+1/(sqrt(7))x^7|+C

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integral of (x^6)/(sqrt(x^{14)-7)}

\int \frac{x ^{6}}{x ^{14} - 7} d x

\frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7} + C

Solution steps

\int \frac{x ^{6}}{x ^{14} - 7} d x

Apply u-substitution

: \int \frac{1}{7 u ^{2} - 7} d u

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

Apply Trigonometric Substitution

: \int sec (v) d v

= \frac{1}{7} \cdot \int sec (v) d v

Use the common integral:

\int sec (v) d v = ln ∣ tan (v) + sec (v) ∣

= \frac{1}{7} ln ∣ tan (v) + sec (v) ∣

Substitute back

= \frac{1}{7} ln tan (arcsec (\frac{1}{7} x^{7})) + sec (arcsec (\frac{1}{7} x^{7}))

Simplify

\frac{1}{7} ln tan (arcsec (\frac{1}{7} x^{7})) + sec (arcsec (\frac{1}{7} x^{7})) : \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7}

= \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7}

Add a constant to the solution

= \frac{1}{7} ln \frac{1}{7} (x^{14} - 7) + \frac{1}{7} x^{7} + C

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What is the integral of (x^6)/(sqrt(x^{14)-7)} ?
The integral of (x^6)/(sqrt(x^{14)-7)} is 1/7 ln|sqrt(1/7 (x^{14)-7)}+1/(sqrt(7))x^7|+C