What is the integral of ((a+bx^7))/(sqrt(8ax+bx^8)) ?

The integral of ((a+bx^7))/(sqrt(8ax+bx^8)) is 1/4 sqrt(8ax+bx^8)+C

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integral of ((a+bx^7))/(sqrt(8ax+bx^8))

\int \frac{( a + b x ^{7} )}{8 a x + b x ^{8}} d x

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

Solution steps

\int \frac{a + b x ^{7}}{8 a x + b x ^{8}} 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 = 8 a x + b x^{8}

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

Simplify

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

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

Add a constant to the solution

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

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