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受欢迎的微积分 >

积分 (sin^2(x))/((1+sin(x)))

解答

积分

\frac{sin ^{2} ( x )}{( 1 + sin ( x ) )}

解答

- \frac{2}{tan ( \frac{x}{2} ) + 1} - cos (x) - x + C

求解步骤

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

使用三角恒等式改写

= \int \frac{1 - cos ^{2} ( x )}{1 + sin ( x )} d x

乘开

\frac{1 - cos ^{2} ( x )}{1 + sin ( x )} : \frac{1}{1 + sin ( x )} - \frac{cos ^{2} ( x )}{1 + sin ( x )}

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

= \int \frac{1}{1 + sin ( x )} d x - \int \frac{cos ^{2} ( x )}{1 + sin ( x )} d x

\int \frac{1}{1 + sin ( x )} d x = - \frac{2}{tan ( \frac{x}{2} ) + 1}

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

= - \frac{2}{tan ( \frac{x}{2} ) + 1} - (cos (x) + x)

化简

= - \frac{2}{tan ( \frac{x}{2} ) + 1} - cos (x) - x

解答补常数

= - \frac{2}{tan ( \frac{x}{2} ) + 1} - cos (x) - x + C

作图

流行的例子

积分 2/(x^{1/3)}

in t e g r a l \frac{2}{x ^{\frac{1}{3}}}

积分 cos^2(t)sin(t)

in t e g r a l cos^{2} (t) sin (t)

积分 1/(36+x^2)

in t e g r a l \frac{1}{36 + x ^{2}}

积分 ((1+x)^2)/((x(1+x^2)))

in t e g r a l \frac{( 1 + x ) ^{2}}{( x ( 1 + x ^{2} ) )}

积分 x/((4+2x))

in t e g r a l \frac{x}{( 4 + 2 x )}