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人気のある微分積分 >

integral of (x+8)/(x^2+x+2)

解

\int \frac{x + 8}{x ^{2} + x + 2} d x

解

\frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{15}{7} arctan (\frac{2 x + 1}{7}) + C

解答ステップ

\int \frac{x + 8}{x ^{2} + x + 2} d x

u置換積分法を適用する

= \int \frac{u}{u ^{2} - 15 u + 58} d u

平方完成する

u^{2} - 15 u + 58 : (u - \frac{15}{2})^{2} + \frac{7}{4}

= \int \frac{u}{( u - \frac{15}{2} ) ^{2} + \frac{7}{4}} d u

u置換積分法を適用する

= \int \frac{2 ( 2 v + 15 )}{4 v ^{2} + 7} d v

定数を除く:

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

= 2 \cdot \int \frac{2 v + 15}{4 v ^{2} + 7} d v

拡張

\frac{2 v + 15}{4 v ^{2} + 7} : \frac{2 v}{4 v ^{2} + 7} + \frac{15}{4 v ^{2} + 7}

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

= 2 (\int \frac{2 v}{4 v ^{2} + 7} d v + \int \frac{15}{4 v ^{2} + 7} d v)

\int \frac{2 v}{4 v ^{2} + 7} d v = \frac{1}{4} ln 4 v^{2} + 7

\int \frac{15}{4 v ^{2} + 7} d v = \frac{15}{2 7} arctan (\frac{2}{7} v)

= 2 (\frac{1}{4} ln 4 v^{2} + 7 + \frac{15}{2 7} arctan (\frac{2}{7} v))

代用を戻す

= 2 (\frac{1}{4} ln 4 (x + 8 - \frac{15}{2})^{2} + 7 + \frac{15}{2 7} arctan (\frac{2}{7} (x + 8 - \frac{15}{2})))

簡素化

2 (\frac{1}{4} ln 4 (x + 8 - \frac{15}{2})^{2} + 7 + \frac{15}{2 7} arctan (\frac{2}{7} (x + 8 - \frac{15}{2}))) : \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{15}{7} arctan (\frac{2 x + 1}{7})

= \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{15}{7} arctan (\frac{2 x + 1}{7})

定数を解答に追加する

= \frac{1}{2} ln 4 x^{2} + 4 x + 8 + \frac{15}{7} arctan (\frac{2 x + 1}{7}) + C

グラフ

人気の例

integral of (x^2)/((7+x^3)^2)

\int \frac{x ^{2}}{( 7 + x ^{3} ) ^{2}} d x

integral of (x^3+4)/(x^5+14x^3+40x)

\int \frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} d x

integral of 1/(x^2-8x+17)

\int \frac{1}{x ^{2} - 8 x + 17} d x

integral of cos(x)sin(x)x

\int cos (x) sin (x) x d x

integral of (log_{10}(x))^2

\int (lo g_{10} (x))^{2} d x