The integral of sec theta with respect to theta is ln|sec theta + tan theta| + C, where C is the constant of integration.
Here's a breakdown of the solution:
- Secant function: Sec theta is the reciprocal of the cosine function, meaning sec theta = 1/cos theta.
- Integration by substitution: We can solve this integral using a substitution method. Let's substitute u = sec theta + tan theta. Then, du = (sec theta tan theta + sec^2 theta) dtheta.
- Simplifying the integral: Notice that the numerator of du is already present in the integrand. We can rewrite the integral in terms of u: ∫ sec theta dtheta = ∫ (1/u) du.
- Integrating: The integral of 1/u is ln|u| + C.
- Substituting back: Substituting back u = sec theta + tan theta, we get the final answer: ln|sec theta + tan theta| + C.
Practical Insights:
- This integral is commonly used in various fields, including physics, engineering, and mathematics.
- It's important to remember the absolute value in the natural logarithm, as the secant and tangent functions can be negative.
- The constant of integration, C, represents the family of all possible antiderivatives.
Example:
Let's find the integral of sec theta from 0 to π/4:
- ∫(from 0 to π/4) sec theta dtheta = ln|sec(π/4) + tan(π/4)| - ln|sec(0) + tan(0)|
- Simplifying, we get: ln|√2 + 1| - ln|1 + 0| = ln(√2 + 1).
Therefore, the definite integral of sec theta from 0 to π/4 is ln(√2 + 1).
