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人気のある微分積分 >

integral of (csc(4x)+cot(4x))^2

解

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

解

\frac{1}{2} (- cot (2 x) - arctan (tan (2 x))) + C

解答ステップ

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

u置換積分法を適用する

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

定数を除く:

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

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

u置換積分法を適用する

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

以下の部分分数を得る:

\frac{1}{v ^{2} ( 1 + v ^{2} )} : \frac{1}{v ^{2}} - \frac{1}{v ^{2} + 1}

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

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}{4} \cdot 2 (\int \frac{1}{v ^{2}} d v - \int \frac{1}{v ^{2} + 1} d v)

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

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

= \frac{1}{4} \cdot 2 (- \frac{1}{v} - arctan (v))

代用を戻す

= \frac{1}{4} \cdot 2 (- \frac{1}{tan ( \frac{4 x}{2} )} - arctan (tan (\frac{4 x}{2})))

簡素化

\frac{1}{4} \cdot 2 (- \frac{1}{tan ( \frac{4 x}{2} )} - arctan (tan (\frac{4 x}{2}))) : \frac{1}{2} (- cot (2 x) - arctan (tan (2 x)))

= \frac{1}{2} (- cot (2 x) - arctan (tan (2 x)))

定数を解答に追加する

= \frac{1}{2} (- cot (2 x) - arctan (tan (2 x))) + C

グラフ

人気の例

integral of e^{3x}*cos(5x)

\int e^{3 x} \cdot cos (5 x) d x

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\int \frac{6 + sin ( x )}{cos ( x )} d x

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\int e^{2 x} \cdot cos (6 x) d x

integral of (x^2-2)cos(2x)

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