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受欢迎的微积分 >

integral of x/(1+sin(x))

解答

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

解答

x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ + C

求解步骤

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

乘以

1 + sin (x)

的共轭根式

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

= \int \frac{x}{\frac{c o s ^{2} ( x )}{1 - s i n ( x )}} d x

乘开

\frac{x}{\frac{c o s ^{2} ( x )}{1 - s i n ( x )}} : \frac{x}{cos ^{2} ( x )} - \frac{x sin ( x )}{cos ^{2} ( x )}

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

\int \frac{x}{cos ^{2} ( x )} d x = x tan (x) + ln ∣ cos (x) ∣

\int \frac{x sin ( x )}{cos ^{2} ( x )} d x = x sin (x) tan (x) + x cos (x) - ln ∣ tan (x) + sec (x) ∣

= x tan (x) + ln ∣ cos (x) ∣ - (x sin (x) tan (x) + x cos (x) - ln ∣ tan (x) + sec (x) ∣)

化简

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣

解答补常数

= x tan (x) + ln ∣ cos (x) ∣ - x sin (x) tan (x) - x cos (x) + ln ∣ tan (x) + sec (x) ∣ + C

作图

流行的例子

tangent f(x)= 1/5 x^{10}-1/3 x^6-x+9

t an g e n t f (x) = \frac{1}{5} x^{10} - \frac{1}{3} x^{6} - x + 9

integral of (2y)/((2+3y^2)^3)

\int \frac{2 y}{( 2 + 3 y ^{2} ) ^{3}} d y

(dy)/(dt)=y-y^2

\frac{d y}{d t} = y - y^{2}

limit as x approaches-1 of-x^3+x^2-2

x \to - 1 lim (- x^{3} + x^{2} - 2)

integral of \sqrt[6]{x^5}

\int 6 x^{5} d x