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人気のある微分積分 >

integral of 6/((x^2+1)(x^2+4))

解

\int \frac{6}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

解

2 arctan (x) - arctan (\frac{x}{2}) + C

解答ステップ

\int \frac{6}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

定数を除く:

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

= 6 \cdot \int \frac{1}{( x ^{2} + 1 ) ( x ^{2} + 4 )} d x

以下の部分分数を得る:

\frac{1}{( x ^{2} + 1 ) ( x ^{2} + 4 )} : \frac{1}{3 ( x ^{2} + 1 )} - \frac{1}{3 ( x ^{2} + 4 )}

= 6 \cdot \int \frac{1}{3 ( x ^{2} + 1 )} - \frac{1}{3 ( x ^{2} + 4 )} 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

= 6 (\int \frac{1}{3 ( x ^{2} + 1 )} d x - \int \frac{1}{3 ( x ^{2} + 4 )} d x)

\int \frac{1}{3 ( x ^{2} + 1 )} d x = \frac{1}{3} arctan (x)

\int \frac{1}{3 ( x ^{2} + 4 )} d x = \frac{1}{6} arctan (\frac{x}{2})

= 6 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2}))

簡素化

6 (\frac{1}{3} arctan (x) - \frac{1}{6} arctan (\frac{x}{2})) : 2 arctan (x) - arctan (\frac{x}{2})

= 2 arctan (x) - arctan (\frac{x}{2})

定数を解答に追加する

= 2 arctan (x) - arctan (\frac{x}{2}) + C

グラフ

人気の例

integral of (6x^3)/(x^4+8x^2+16)

\int \frac{6 x ^{3}}{x ^{4} + 8 x ^{2} + 16} d x

integral of 2/((3x+5)^2)

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

integral of cot^{10}(x)csc^4(x)

\int cot^{10} (x) csc^{4} (x) d x

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

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

integral of ln(4)4^x

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