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受欢迎的微积分 >

integral of 1/(cos^4(3x))

解答

\int \frac{1}{cos ^{4} ( 3 x )} d x

解答

\frac{1}{3} (tan (3 x) + \frac{tan ^{3} ( 3 x )}{3}) + C

求解步骤

\int \frac{1}{cos ^{4} ( 3 x )} d x

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{4} (3 x) d x

使用换元积分法

= \int sec^{4} (u) \frac{1}{3} 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 sec^{4} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int (1 + tan^{2} (u)) sec^{2} (u) d u

使用换元积分法

= \frac{1}{3} \cdot \int 1 + v^{2} d v

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 1 d v + \int v^{2} d v)

\int 1 d v = v

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{1}{3} (v + \frac{v ^{3}}{3})

代回

= \frac{1}{3} (tan (3 x) + \frac{tan ^{3} ( 3 x )}{3})

解答补常数

= \frac{1}{3} (tan (3 x) + \frac{tan ^{3} ( 3 x )}{3}) + C

作图

流行的例子

integral of tan(x)ln(sec(x))

\int tan (x) ln (sec (x)) d x

integral of (2x)/((7x^2+3)^5)

\int \frac{2 x}{( 7 x ^{2} + 3 ) ^{5}} d x

integral of 3sqrt(4-25s^2)

\int 3 4 - 25 s^{2} d s

integral of 8/(cos^4(x))

\int \frac{8}{cos ^{4} ( x )} d x

integral of 1/(x\sqrt[3]{ln(x^5))}

\int \frac{1}{x 3 ln ( x ^{5} )} d x