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受欢迎的微积分 >

integral of sqrt((sin(x))/(cos(x)))

解答

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

解答

22 (- \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{sin ( x )}{cos ( x )} d x

使用基本三角恒等式:

\frac{sin ( x )}{cos ( x )} = tan (x)

= \int tan (x) d x

使用三角换元法

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

使用换元积分法

= \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

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

使用换元积分法

= 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

= 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 )}

= 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

= 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))

= 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

= 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

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 (- \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 (- \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 (- \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

流行的例子

integral of (xsqrt(8-x^2))

\int (x 8 - x^{2}) d x

integral of (2x+3)/(x^2+4x+10)

\int \frac{2 x + 3}{x ^{2} + 4 x + 10} d x

integral of pi(sec^2(x))(tan(x)+4)^4

\int π (sec^{2} (x)) (tan (x) + 4)^{4} d x

integral of e^xcos(0.5x)

\int e^{x} cos (0.5 x) d x

integral of (x^p)/(p+1)

\int \frac{x ^{p}}{p + 1} d x