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

解答

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

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

求解步骤

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

使用指数法则:

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

a x^{4} = x^{\frac{4}{a}}

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

使用幂法则

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

带入上下限计算后得:

\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}

化简

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

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