The antiderivative of sec x tan x is closely related to the trigonometric functions secant and tangent, as well as their derivatives and integrals. The antiderivative itself, denoted as sec x, represents the area under the curve of the sec x tan x function, while its derivative is sec^2 x, the square of the secant function. The integral of sec x tan x, on the other hand, is ln(sec x + tan x) + C, where C is the constant of integration.
How to Find the Antiderivative of sec x tan x
The antiderivative of sec x tan x is sec x. This can be proven using the following steps:
- Let u = sec x. Then du/dx = sec x tan x.
- Integrate both sides of the equation with respect to x:
∫du = ∫sec x tan x dx
u = sec x + C
- Substitute u back into the original equation:
sec x = sec x + C
- Solve for C:
C = 0
Therefore, the antiderivative of sec x tan x is sec x.
Here is a table summarizing the steps:
| Step | Equation |
|---|---|
| 1 | Let u = sec x. Then du/dx = sec x tan x. |
| 2 | Integrate both sides of the equation with respect to x: ∫du = ∫sec x tan x dx |
| 3 | Substitute u back into the original equation: u = sec x + C |
| 4 | Solve for C: C = 0 |
You can also use the following bullet points to remember the steps:
- Let u = sec x.
- Integrate both sides of the equation with respect to x.
- Substitute u back into the original equation.
- Solve for C.
Question 1:
What is the technique used to determine the antiderivative of sec x tan x?
Answer:
The technique used to determine the antiderivative of sec x tan x is substitution, specifically with u = sec x. This substitution allows the integral to be rewritten as (1/u) du, which has an antiderivative of ln |u|. Substituting back for u, the antiderivative of sec x tan x is ln |sec x|.
Question 2:
How does the chain rule apply to the integration of sec x tan x?
Answer:
The chain rule is used to determine the derivative of sec x tan x, which is sec^2 x tan x. When integrating sec x tan x, the antiderivative of sec^2 x tan x is found by substituting u = sec x and applying the chain rule with du/dx = sec x tan x. The antiderivative of sec^2 x tan x is then (1/sec x tan x) du, which is ln |sec x|.
Question 3:
What is the geometric interpretation of the antiderivative of sec x tan x?
Answer:
The antiderivative of sec x tan x, ln |sec x|, is the natural logarithmic function of the secant of x. The secant of x is the length of the hypotenuse of a right triangle with an angle of x and opposite side length of 1. The natural logarithm of the secant of x is the area of the sector of a circle with radius 1 and central angle of x.
That’s all there is to it! Now you know how to find the antiderivative of sec x tan x. Thanks for sticking with me through this math adventure. If you have any more calculus questions, be sure to come back and visit. I’m always happy to help.