integral of 1/(cos(4θ))

解答

\int \frac{1}{cos ( 4 θ )} d θ

\frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣ + C

求解步骤

\int \frac{1}{cos ( 4 θ )} d θ

使用三角恒等式改写

= \int sec (4 θ) d θ

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{4} ln ∣ tan (u) + sec (u) ∣

u = 4 θ

代回

= \frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣

解答补常数

= \frac{1}{4} ln ∣ tan (4 θ) + sec (4 θ) ∣ + C

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