What is the integral from-6 to 6 of \sqrt[a]{x^4} ?

The integral from-6 to 6 of \sqrt[a]{x^4} is (6^{(4+a)/a}a-(-6)^{(4+a)/a}a)/(4+a)

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integral from-6 to 6 of \sqrt[a]{x^4}

\frac{6 ^{\frac{4 + a}{a}} a - ( - 6 ) ^{\frac{4 + a}{a}} a}{4 + a}

Solution steps

Apply exponent rule:

(a^{n})^{m} = a^{n \cdot m}

= \int_{- 6}^{6} x^{\frac{4}{a}} d x

Apply the Power Rule

: [\frac{a x ^{\frac{4 + a}{a}}}{4 + a}]_{- 6}^{6}

= [\frac{a x ^{\frac{4 + a}{a}}}{4 + a}]_{- 6}^{6}

Compute the boundaries:

\frac{a \cdot 6 ^{\frac{4 + a}{a}}}{4 + a} - \frac{a ( - 6 ) ^{\frac{4 + a}{a}}}{4 + a}

= \frac{a \cdot 6 ^{\frac{4 + a}{a}}}{4 + a} - \frac{a ( - 6 ) ^{\frac{4 + a}{a}}}{4 + a}

Simplify

= \frac{6 ^{\frac{4 + a}{a}} a - ( - 6 ) ^{\frac{4 + a}{a}} a}{4 + a}

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What is the integral from-6 to 6 of \sqrt[a]{x^4} ?
The integral from-6 to 6 of \sqrt[a]{x^4} is (6^{(4+a)/a}a-(-6)^{(4+a)/a}a)/(4+a)