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人気のある微分積分 >

integral of (6x-5)/(x^2+100)

解

\int \frac{6 x - 5}{x ^{2} + 100} d x

解

\frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10})) + C

解答ステップ

\int \frac{6 x - 5}{x ^{2} + 100} d x

置換積分法を適用する

= \int \frac{12 u - 1}{2 ( u ^{2} + 1 )} 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{12 u - 1}{u ^{2} + 1} d u

拡張

\frac{12 u - 1}{u ^{2} + 1} : \frac{12 u}{u ^{2} + 1} - \frac{1}{u ^{2} + 1}

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int \frac{12 u}{u ^{2} + 1} d u - \int \frac{1}{u ^{2} + 1} d u)

\int \frac{12 u}{u ^{2} + 1} d u = 6 ln u^{2} + 1

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{1}{2} (6 ln u^{2} + 1 - arctan (u))

代用を戻す

u = \frac{x}{10}

= \frac{1}{2} (6 ln (\frac{x}{10})^{2} + 1 - arctan (\frac{x}{10}))

(\frac{x}{10})^{2} = \frac{x ^{2}}{100}

= \frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10}))

定数を解答に追加する

= \frac{1}{2} (6 ln \frac{x ^{2}}{100} + 1 - arctan (\frac{x}{10})) + C

グラフ

人気の例

integral of 1/(x(sqrt(16+x^2)))

\int \frac{1}{x ( 16 + x ^{2} )} d x

integral of (1+2sin(x))^2

\int (1 + 2 sin (x))^{2} d x

integral of (4x^2+x+2)/(x+1)

\int \frac{4 x ^{2} + x + 2}{x + 1} d x

integral of (3x^2+5x-6)

\int (3 x^{2} + 5 x - 6) d x

integral of 1/(sqrt(y)-y)

\int \frac{1}{y - y} d y