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人気のある微分積分 >

integral of (cot(x))/(sqrt(4-csc^2(x)))

解

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

解

\frac{1}{4} (ln \frac{4 - csc ^{2} ( x )}{2} + 1 - ln \frac{4 - csc ^{2} ( x )}{2} - 1) + C

解答ステップ

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

u置換積分法を適用する

= \int - \frac{1}{u 4 - u ^{2}} d u

定数を除く:

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

= - \int \frac{1}{u 4 - u ^{2}} d u

u置換積分法を適用する

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

定数を除く:

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

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

置換積分法を適用する

= - (- \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 \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 = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

= - (- \frac{1}{2} (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}))

代用を戻す

= - - \frac{1}{2} \frac{ln \frac{4 - c s c ^{2} ( x )}{2} + 1}{2} - \frac{ln \frac{4 - c s c ^{2} ( x )}{2} - 1}{2}

簡素化

- - \frac{1}{2} \frac{ln \frac{4 - c s c ^{2} ( x )}{2} + 1}{2} - \frac{ln \frac{4 - c s c ^{2} ( x )}{2} - 1}{2} : \frac{1}{4} (ln \frac{4 - csc ^{2} ( x )}{2} + 1 - ln \frac{4 - csc ^{2} ( x )}{2} - 1)

= \frac{1}{4} (ln \frac{4 - csc ^{2} ( x )}{2} + 1 - ln \frac{4 - csc ^{2} ( x )}{2} - 1)

定数を解答に追加する

= \frac{1}{4} (ln \frac{4 - csc ^{2} ( x )}{2} + 1 - ln \frac{4 - csc ^{2} ( x )}{2} - 1) + C

グラフ

人気の例

integral of (sec(y))/(sec(y)-1)

\int \frac{sec ( y )}{sec ( y ) - 1} d y

integral of sec^2(x)ln(cos(x))

\int sec^{2} (x) ln (cos (x)) d x

integral of (x-5)(x+3)

\int (x - 5) (x + 3) d x

integral of (2x-2)e^{3x}

\int (2 x - 2) e^{3 x} d x

integral of (2x+1)/(\sqrt[4]{x^2+x)}

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