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人気のある微分積分 >

integral of (sin(x)cos(x))/(4+sin^4(x))

解

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

解

\frac{1}{4} arctan (\frac{1}{2} sin^{2} (x)) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{u}{4 + u ^{4}} d u

u置換積分法を適用する

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

置換積分法を適用する

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

共通積分を使用する:

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

= \frac{1}{2} \cdot \frac{1}{2} arctan (w)

代用を戻す

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

簡素化

\frac{1}{2} \cdot \frac{1}{2} arctan (\frac{sin ^{2} ( x )}{2}) : \frac{1}{4} arctan (\frac{1}{2} sin^{2} (x))

= \frac{1}{4} arctan (\frac{1}{2} sin^{2} (x))

定数を解答に追加する

= \frac{1}{4} arctan (\frac{1}{2} sin^{2} (x)) + C

グラフ

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