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人気のある微分積分 >

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

解

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

解

\frac{1}{32} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2}))) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{1}{2 ( 4 + u ^{2} ) ^{2}} 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{1}{( 4 + u ^{2} ) ^{2}} d u

置換積分法を適用する

= \frac{1}{2} \cdot \int \frac{1}{8 ( v ^{2} + 1 ) ^{2}} d v

定数を除く:

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

= \frac{1}{2} \cdot \frac{1}{8} \cdot \int \frac{1}{( v ^{2} + 1 ) ^{2}} d v

三角関数による置換を適用する

= \frac{1}{2} \cdot \frac{1}{8} \cdot \int \frac{1}{sec ^{4} ( w ) cos ^{2} ( w )} d w

三角関数の公式を使用して書き換える

= \frac{1}{2} \cdot \frac{1}{8} \cdot \int \frac{1 + cos ( 2 w )}{2} d w

定数を除く:

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

= \frac{1}{2} \cdot \frac{1}{8} \cdot \frac{1}{2} \cdot \int 1 + cos (2 w) d w

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} \cdot \frac{1}{8} \cdot \frac{1}{2} (\int 1 d w + \int cos (2 w) d w)

\int 1 d w = w

\int cos (2 w) d w = \frac{1}{2} sin (2 w)

= \frac{1}{2} \cdot \frac{1}{8} \cdot \frac{1}{2} (w + \frac{1}{2} sin (2 w))

代用を戻す

= \frac{1}{2} \cdot \frac{1}{8} \cdot \frac{1}{2} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2})))

簡素化

\frac{1}{2} \cdot \frac{1}{8} \cdot \frac{1}{2} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2}))) : \frac{1}{32} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2})))

= \frac{1}{32} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2})))

定数を解答に追加する

= \frac{1}{32} (arctan (\frac{x ^{2}}{2}) + \frac{1}{2} sin (2 arctan (\frac{x ^{2}}{2}))) + C

グラフ

人気の例

derivative of-arctan(x/y)

\frac{d}{d x} (- arctan (\frac{x}{y}))

derivative of sin(x^2-x)

\frac{d}{d x} (sin (x^{2} - x))

tangent y=(1+3x)^8,(0,1)

t an g e n t y = (1 + 3 x)^{8}, (0, 1)

integral of (x^2e^{2x})

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

limit as x approaches-infinity of f(x)

x \to - \infty lim (f (x))