What is the integral of ((a+bx^6))/(sqrt(7ax+bx^7)) ?

The integral of ((a+bx^6))/(sqrt(7ax+bx^7)) is 2/7 sqrt(7ax+bx^7)+C

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integral of ((a+bx^6))/(sqrt(7ax+bx^7))

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

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

Solution steps

\int \frac{a + b x ^{6}}{7 a x + b x ^{7}} d x

Apply u-substitution

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

= \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

: 2 u^{\frac{1}{2}}

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

Substitute back

u = 7 a x + b x^{7}

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

Simplify

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

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

Add a constant to the solution

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

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What is the integral of ((a+bx^6))/(sqrt(7ax+bx^7)) ?
The integral of ((a+bx^6))/(sqrt(7ax+bx^7)) is 2/7 sqrt(7ax+bx^7)+C