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受欢迎的微积分 >

积分 (1+xtan(x)-sin(x))sec^2(x)

解答

积分

(1 + x tan (x) - sin (x)) sec^{2} (x)

解答

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

求解步骤

\int (1 + x tan (x) - sin (x)) sec^{2} (x) d x

使用三角恒等式改写

= \int \frac{cos ( x ) + x sin ( x ) - sin ( x ) cos ( x )}{cos ^{3} ( x )} d x

乘开

\frac{cos ( x ) + x sin ( x ) - sin ( x ) cos ( x )}{cos ^{3} ( x )} : \frac{1}{cos ^{2} ( x )} + \frac{x sin ( x )}{cos ^{3} ( x )} - \frac{sin ( x )}{cos ^{2} ( x )}

= \int \frac{1}{cos ^{2} ( x )} + \frac{x sin ( x )}{cos ^{3} ( x )} - \frac{sin ( x )}{cos ^{2} ( x )} d 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}{cos ^{2} ( x )} d x + \int \frac{x sin ( x )}{cos ^{3} ( x )} d x - \int \frac{sin ( x )}{cos ^{2} ( x )} d x

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

\int \frac{x sin ( x )}{cos ^{3} ( x )} d x = \frac{1}{2} x tan^{2} (x) - \frac{1}{2} (- x + tan (x))

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

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

化简

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

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

解答补常数

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

作图

流行的例子

积分 cos^2(x)-sin^2(x)

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

in t e g r a l 5 + 2 x

积分 ((x-1))/((x+1))

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

积分 1/(2x-1)

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

积分 ((x^2+2x+1))/(x^2)

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