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受欢迎的微积分 >

integral of (2tan(x)+3cot(x))^2

解答

\int (2 tan (x) + 3 cot (x))^{2} d x

解答

\frac{9 tan ^{3} ( \frac{x}{2} ) + 9 cot ( \frac{x}{2} ) + 4 arctan ( tan ( \frac{x}{2} ) ) - 9 tan ^{2} ( \frac{x}{2} ) cot ( \frac{x}{2} ) - 4 tan ^{2} ( \frac{x}{2} ) arctan ( tan ( \frac{x}{2} ) ) - 25 tan ( \frac{x}{2} )}{2 ( tan ( \frac{x}{2} ) + 1 ) ( tan ( \frac{x}{2} ) - 1 )} + C

求解步骤

\int (2 tan (x) + 3 cot (x))^{2} d x

使用换元积分法

= \int \frac{( - 2 u ^{2} - 3 u ^{4} - 3 ) ^{2}}{2 u ^{2} ( u + 1 ) ^{2} ( u - 1 ) ^{2} ( u ^{2} + 1 )} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{( - 2 u ^{2} - 3 u ^{4} - 3 ) ^{2}}{u ^{2} ( u + 1 ) ^{2} ( u - 1 ) ^{2} ( u ^{2} + 1 )} d u

将

\frac{( - 2 u ^{2} - 3 u ^{4} - 3 ) ^{2}}{u ^{2} ( u + 1 ) ^{2} ( u - 1 ) ^{2} ( u ^{2} + 1 )}

用部份分式展开:

9 + \frac{9}{u ^{2}} + \frac{8}{( u + 1 ) ^{2}} + \frac{8}{( u - 1 ) ^{2}} - \frac{4}{u ^{2} + 1}

= \frac{1}{2} \cdot \int 9 + \frac{9}{u ^{2}} + \frac{8}{( u + 1 ) ^{2}} + \frac{8}{( u - 1 ) ^{2}} - \frac{4}{u ^{2} + 1} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int 9 d u + \int \frac{9}{u ^{2}} d u + \int \frac{8}{( u + 1 ) ^{2}} d u + \int \frac{8}{( u - 1 ) ^{2}} d u - \int \frac{4}{u ^{2} + 1} d u)

\int 9 d u = 9 u

\int \frac{9}{u ^{2}} d u = - \frac{9}{u}

\int \frac{8}{( u + 1 ) ^{2}} d u = - \frac{8}{u + 1}

\int \frac{8}{( u - 1 ) ^{2}} d u = - \frac{8}{u - 1}

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

= \frac{1}{2} (9 u - \frac{9}{u} - \frac{8}{u + 1} - \frac{8}{u - 1} - 4 arctan (u))

u = tan (\frac{x}{2})

代回

= \frac{1}{2} (9 tan (\frac{x}{2}) - \frac{9}{tan ( \frac{x}{2} )} - \frac{8}{tan ( \frac{x}{2} ) + 1} - \frac{8}{tan ( \frac{x}{2} ) - 1} - 4 arctan (tan (\frac{x}{2})))

化简

\frac{1}{2} (9 tan (\frac{x}{2}) - \frac{9}{tan ( \frac{x}{2} )} - \frac{8}{tan ( \frac{x}{2} ) + 1} - \frac{8}{tan ( \frac{x}{2} ) - 1} - 4 arctan (tan (\frac{x}{2}))) : \frac{9 tan ^{3} ( \frac{x}{2} ) + 9 cot ( \frac{x}{2} ) + 4 arctan ( tan ( \frac{x}{2} ) ) - 9 tan ^{2} ( \frac{x}{2} ) cot ( \frac{x}{2} ) - 4 tan ^{2} ( \frac{x}{2} ) arctan ( tan ( \frac{x}{2} ) ) - 25 tan ( \frac{x}{2} )}{2 ( tan ( \frac{x}{2} ) + 1 ) ( tan ( \frac{x}{2} ) - 1 )}

= \frac{9 tan ^{3} ( \frac{x}{2} ) + 9 cot ( \frac{x}{2} ) + 4 arctan ( tan ( \frac{x}{2} ) ) - 9 tan ^{2} ( \frac{x}{2} ) cot ( \frac{x}{2} ) - 4 tan ^{2} ( \frac{x}{2} ) arctan ( tan ( \frac{x}{2} ) ) - 25 tan ( \frac{x}{2} )}{2 ( tan ( \frac{x}{2} ) + 1 ) ( tan ( \frac{x}{2} ) - 1 )}

解答补常数

= \frac{9 tan ^{3} ( \frac{x}{2} ) + 9 cot ( \frac{x}{2} ) + 4 arctan ( tan ( \frac{x}{2} ) ) - 9 tan ^{2} ( \frac{x}{2} ) cot ( \frac{x}{2} ) - 4 tan ^{2} ( \frac{x}{2} ) arctan ( tan ( \frac{x}{2} ) ) - 25 tan ( \frac{x}{2} )}{2 ( tan ( \frac{x}{2} ) + 1 ) ( tan ( \frac{x}{2} ) - 1 )} + C

作图

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