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人気のある微分積分 >

integral of 1/(xsqrt(a^2+x^2))

解

\int \frac{1}{x a ^{2} + x ^{2}} d x

解

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

解答ステップ

\int \frac{1}{x a ^{2} + x ^{2}} d x

u置換積分法を適用する

= \int \frac{1}{2 u a ^{2} + u} d u

定数を除く:

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

= \frac{1}{2} \cdot \int \frac{1}{u a ^{2} + u} d u

u置換積分法を適用する

= \frac{1}{2} \cdot \int \frac{2}{v ^{2} - a ^{2}} d v

定数を除く:

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

= \frac{1}{2} \cdot 2 \cdot \int \frac{1}{v ^{2} - a ^{2}} d v

置換積分法を適用する

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

定数を除く:

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

= \frac{1}{2} \cdot 2 \cdot \frac{1}{a} \cdot \int \frac{1}{w ^{2} - 1} d w

因数

w^{2} - 1 : - (- w^{2} + 1)

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

定数を除く:

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

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

共通積分を使用する:

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= \frac{1}{2} \cdot 2 \cdot \frac{1}{a} (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代用を戻す

= \frac{1}{2} \cdot 2 \cdot \frac{1}{a} - \frac{ln \frac{a ^{2} + x ^{2}}{a} + 1}{2} - \frac{ln \frac{a ^{2} + x ^{2}}{a} - 1}{2}

簡素化

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

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

定数を解答に追加する

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

グラフ

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