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人気のある微分積分 >

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

解

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

解

\frac{1}{4} (2 x^{2} + ln ∣ x ∣) + C

解答ステップ

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

簡素化

1 + (x - \frac{1}{4 x})^{2} : \frac{4 x ^{2} + 1}{4 x}

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

定数を除く:

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

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

拡張

\frac{4 x ^{2} + 1}{x} : 4 x + \frac{1}{x}

= \frac{1}{4} \cdot \int 4 x + \frac{1}{x} d x

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}{4} (\int 4 x d x + \int \frac{1}{x} d x)

\int 4 x d x = 2 x^{2}

\int \frac{1}{x} d x = ln ∣ x ∣

= \frac{1}{4} (2 x^{2} + ln ∣ x ∣)

定数を解答に追加する

= \frac{1}{4} (2 x^{2} + ln ∣ x ∣) + C

グラフ

人気の例

integral of 1/(x^{1.01)}

\int \frac{1}{x ^{1.01}} d x

integral of 25z^2(125z^3+1)^{12}

\int 25 z^{2} (125 z^{3} + 1)^{12} d z

integral of (e^x-cos(x))^2

\int (e^{x} - cos (x))^{2} d x

integral of 2/(100-x)

\int \frac{2}{100 - x} d x

integral of (x^3+2x^2)/x

\int \frac{x ^{3} + 2 x ^{2}}{x} d x