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人気のある微分積分 >

integral of 1/(sin^4(x)cos^2(x))

解

\int \frac{1}{sin ^{4} ( x ) cos ^{2} ( x )} d x

解

- (- tan (x) + 2 cot (x) + \frac{1}{3} cot^{3} (x)) + C

解答ステップ

\int \frac{1}{sin ^{4} ( x ) cos ^{2} ( x )} d x

三角関数による置換を適用する

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

拡張

\frac{( 1 + u ^{2} ) ^{2}}{u ^{2}} : \frac{1}{u ^{2}} + 2 + u^{2}

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

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

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

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

\int 2 d u = 2 u

\int u^{2} d u = \frac{u ^{3}}{3}

= - (- \frac{1}{u} + 2 u + \frac{u ^{3}}{3})

代用を戻す

u = cot (x)

= - (- \frac{1}{cot ( x )} + 2 cot (x) + \frac{cot ^{3} ( x )}{3})

簡素化

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

= - (- tan (x) + 2 cot (x) + \frac{1}{3} cot^{3} (x))

定数を解答に追加する

= - (- tan (x) + 2 cot (x) + \frac{1}{3} cot^{3} (x)) + C

グラフ

人気の例

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