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人気のある微分積分 >

integral of (x^2)/(3x-1)

解

\int \frac{x ^{2}}{3 x - 1} d x

解

\frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) + C

解答ステップ

\int \frac{x ^{2}}{3 x - 1} d x

u置換積分法を適用する

= \int \frac{( u + 1 ) ^{2}}{27 u} d u

定数を除く:

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

= \frac{1}{27} \cdot \int \frac{( u + 1 ) ^{2}}{u} d u

拡張

\frac{( u + 1 ) ^{2}}{u} : u + 2 + \frac{1}{u}

= \frac{1}{27} \cdot \int u + 2 + \frac{1}{u} d u

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}{27} (\int u d u + \int 2 d u + \int \frac{1}{u} d u)

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= \frac{1}{27} (\frac{u ^{2}}{2} + 2 u + ln ∣ u ∣)

代用を戻す

u = 3 x - 1

= \frac{1}{27} (\frac{( 3 x - 1 ) ^{2}}{2} + 2 (3 x - 1) + ln ∣ 3 x - 1 ∣)

簡素化

\frac{1}{27} (\frac{( 3 x - 1 ) ^{2}}{2} + 2 (3 x - 1) + ln ∣ 3 x - 1 ∣) : \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3)

= \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3)

定数を解答に追加する

= \frac{1}{54} (9 x^{2} + 6 x + 2 ln ∣ 3 x - 1 ∣ - 3) + C

グラフ

人気の例

integral of (sin(x/2))/2

\int \frac{sin ( \frac{x}{2} )}{2} d x

integral of 1/(t^2+3)

\int \frac{1}{t ^{2} + 3} d t

integral of ((x^3-1)/x)^2

\int (\frac{x ^{3} - 1}{x})^{2} d x

integral of (4(ln(x))^2)/(x^2)

\int \frac{4 ( ln ( x ) ) ^{2}}{x ^{2}} d x

integral of-4/(x^4)

\int - \frac{4}{x ^{4}} d x