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受欢迎的微积分 >

integral of (sin(3t))/(cos(3t-1))

解答

\int \frac{sin ( 3 t )}{cos ( 3 t - 1 )} d t

解答

\frac{1}{3} (sin (1) (3 t - 1) - cos (1) ln ∣ cos (3 t - 1) ∣) + C

求解步骤

\int \frac{sin ( 3 t )}{cos ( 3 t - 1 )} d t

使用换元积分法

= \int \frac{sin ( u + 1 )}{3 cos ( u )} d u

提出常数:

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

= \frac{1}{3} \cdot \int \frac{sin ( u + 1 )}{cos ( u )} d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int \frac{cos ( u ) sin ( 1 ) + cos ( 1 ) sin ( u )}{cos ( u )} d u

乘开

\frac{cos ( u ) sin ( 1 ) + cos ( 1 ) sin ( u )}{cos ( u )} : sin (1) + \frac{cos ( 1 ) sin ( u )}{cos ( u )}

= \frac{1}{3} \cdot \int sin (1) + \frac{cos ( 1 ) sin ( u )}{cos ( u )} d u

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

= \frac{1}{3} (\int sin (1) d u + \int \frac{cos ( 1 ) sin ( u )}{cos ( u )} d u)

\int sin (1) d u = sin (1) u

\int \frac{cos ( 1 ) sin ( u )}{cos ( u )} d u = - cos (1) ln ∣ cos (u) ∣

= \frac{1}{3} (sin (1) u - cos (1) ln ∣ cos (u) ∣)

u = 3 t - 1

代回

= \frac{1}{3} (sin (1) (3 t - 1) - cos (1) ln ∣ cos (3 t - 1) ∣)

解答补常数

= \frac{1}{3} (sin (1) (3 t - 1) - cos (1) ln ∣ cos (3 t - 1) ∣) + C

作图

流行的例子

integral of 2x^2-2x+2

\int 2 x^{2} - 2 x + 2 d x

integral of (sqrt(2x^2-16))/(3x)

\int \frac{2 x ^{2} - 16}{3 x} d x

integral of (x^7)/(x^8+1)

\int \frac{x ^{7}}{x ^{8} + 1} d x

integral of x^2-x+3

\int x^{2} - x + 3 d x

integral of 1/(-4+x^2)

\int \frac{1}{- 4 + x ^{2}} d x