integral of 1/(x^n(1+x^n)^{1/n)}

解答

\int \frac{1}{x ^{n} ( 1 + x ^{n} ) ^{\frac{1}{n}}} d x

\frac{x ^{- n + 1}}{- n + 1} + C

求解步骤

\int \frac{1}{x ^{n} ( 1 + x ^{n} ) ^{\frac{1}{n}}} d x

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

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

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

化简

= \int x^{- n} d x

使用幂法则

= \frac{x ^{- n + 1}}{- n + 1}

解答补常数

= \frac{x ^{- n + 1}}{- n + 1} + C

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