What is the integral of (a^2*x)^{((1-a))/a} ?

The integral of (a^2*x)^{((1-a))/a} is a^{(2(1-a))/a+1}x^{1/a}+C

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integral of (a^2*x)^{((1-a))/a}

\int (a^{2} \cdot x)^{\frac{( 1 - a )}{a}} d x

a^{\frac{2 ( 1 - a )}{a} + 1} x^{\frac{1}{a}} + C

Solution steps

\int (a^{2} x)^{\frac{1 - a}{a}} d x

(a^{2} x)^{\frac{1 - a}{a}} = (a^{2})^{\frac{1 - a}{a}} x^{\frac{1 - a}{a}}

= \int (a^{2})^{\frac{1 - a}{a}} x^{\frac{1 - a}{a}} d x

Take the constant out:

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

= (a^{2})^{\frac{1 - a}{a}} \cdot \int x^{\frac{1 - a}{a}} d x

Apply the Power Rule

: a x^{\frac{1}{a}}

= (a^{2})^{\frac{1 - a}{a}} a x^{\frac{1}{a}}

Simplify

(a^{2})^{\frac{1 - a}{a}} a x^{\frac{1}{a}} : a^{\frac{2 ( 1 - a )}{a} + 1} x^{\frac{1}{a}}

= a^{\frac{2 ( 1 - a )}{a} + 1} x^{\frac{1}{a}}

Add a constant to the solution

= a^{\frac{2 ( 1 - a )}{a} + 1} x^{\frac{1}{a}} + C

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What is the integral of (a^2*x)^{((1-a))/a} ?
The integral of (a^2*x)^{((1-a))/a} is a^{(2(1-a))/a+1}x^{1/a}+C