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受欢迎的微积分 >

integral of 1 sqrt(tan(x))

解答

\int \frac{d}{1} tan (x) d x

解答

22 d (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1))) + C

求解步骤

\int \frac{d}{1} tan (x) d x

提出常数:

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

= d \cdot \int tan (x) d x

使用三角换元法

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

使用换元积分法

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

提出常数:

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

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

使用换元积分法

= d 2 \cdot \int \frac{2 w ^{2}}{4 + w ^{4}} d w

提出常数:

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

= d 2 \cdot 2 \cdot \int \frac{w ^{2}}{4 + w ^{4}} d w

将

\frac{w ^{2}}{4 + w ^{4}}

用部份分式展开:

- \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )}

= d 2 \cdot 2 \cdot \int - \frac{w}{4 ( w ^{2} + 2 w + 2 )} + \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w

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

= d 2 \cdot 2 (- \int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w + \int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w)

\int \frac{w}{4 ( w ^{2} + 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1))

\int \frac{w}{4 ( w ^{2} - 2 w + 2 )} d w = \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1))

= d 2 \cdot 2 (- \frac{1}{8} (ln w^{2} + 2 w + 2 - 2 arctan (w + 1)) + \frac{1}{8} (ln w^{2} - 2 w + 2 + 2 arctan (w - 1)))

代回

= d 2 \cdot 2 (- \frac{1}{8} (ln (2 tan (x))^{2} + 2 2 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln (2 tan (x))^{2} - 2 2 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

化简

d 2 \cdot 2 (- \frac{1}{8} (ln (2 tan (x))^{2} + 2 2 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln (2 tan (x))^{2} - 2 2 tan (x) + 2 + 2 arctan (2 tan (x) - 1))) : 22 d (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

= 22 d (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1)))

解答补常数

= 22 d (- \frac{1}{8} (ln 2 tan (x) + 22 tan (x) + 2 - 2 arctan (2 tan (x) + 1)) + \frac{1}{8} (ln 2 tan (x) - 22 tan (x) + 2 + 2 arctan (2 tan (x) - 1))) + C

作图

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\int_{0}^{\frac{3 π}{2}} ∣ sin (x) ∣ d x

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integral of sec(5x-3)tan(5x-3)

\int sec (5 x - 3) tan (5 x - 3) d x

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x \to - 5 lim (- 8)

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\frac{d}{d x} (x^{2} cos (4 x + 3))