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人気のある微分積分 >

integral of 1/(xsqrt(7-x^2))

解

\int \frac{1}{x 7 - x ^{2}} d x

解

- \frac{1}{2 7} (ln \frac{7 - x ^{2}}{7} + 1 - ln \frac{7 - x ^{2}}{7} - 1) + C

解答ステップ

\int \frac{1}{x 7 - x ^{2}} d x

u置換積分法を適用する

= \int - \frac{1}{- u ^{2} + 7} d u

定数を除く:

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

= - \int \frac{1}{- u ^{2} + 7} d u

置換積分法を適用する

= - \int \frac{1}{7 ( - v ^{2} + 1 )} d v

定数を除く:

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

= - \frac{1}{7} \cdot \int \frac{1}{- v ^{2} + 1} d v

共通積分を使用する:

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

= - \frac{1}{7} (\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2})

代用を戻す

= - \frac{1}{7} \frac{ln \frac{7 - x ^{2}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{2}}{7} - 1}{2}

簡素化

- \frac{1}{7} \frac{ln \frac{7 - x ^{2}}{7} + 1}{2} - \frac{ln \frac{7 - x ^{2}}{7} - 1}{2} : - \frac{1}{2 7} (ln \frac{7 - x ^{2}}{7} + 1 - ln \frac{7 - x ^{2}}{7} - 1)

= - \frac{1}{2 7} (ln \frac{7 - x ^{2}}{7} + 1 - ln \frac{7 - x ^{2}}{7} - 1)

定数を解答に追加する

= - \frac{1}{2 7} (ln \frac{7 - x ^{2}}{7} + 1 - ln \frac{7 - x ^{2}}{7} - 1) + C

グラフ

人気の例

integral of 4x-2

\int 4 x - 2 d x

derivative y=2Cx+C(sin(x)+xcos(x))

d er i v a t i v e y = 2 C x + C (sin (x) + x cos (x))

sum from n=1 to infinity of (7/3)^n

n = 1 \sum \infty (\frac{7}{3})^{n}

derivative f(x)=ln(11x)

d er i v a t i v e f (x) = ln (11 x)

(\partial)/(\partial x)(x^2(1+y^2))

\frac{\partial}{\partial x} (x^{2} (1 + y^{2}))