The integral of secx is ln|secx + tanx| + C, where C is the constant of integration.
Understanding the Integral
The integral of a function represents the area under its curve. Finding the integral of secx involves a clever trick using trigonometric identities:
- Multiply by (secx + tanx)/(secx + tanx): This is equivalent to multiplying by 1, so we don't change the expression.
- Simplify: The numerator becomes sec²x + secx tanx, and the denominator becomes secx + tanx.
- Recognize the derivative: The numerator is the derivative of the denominator (d/dx(secx + tanx) = sec²x + secx tanx).
- Integrate: We now have the integral of (d/dx(secx + tanx))/(secx + tanx), which is simply ln|secx + tanx| + C.
Practical Insight
This integral is particularly useful in solving problems related to:
- Arc length: Calculating the length of a curve defined by a function.
- Surface area: Finding the surface area of a solid generated by rotating a curve around an axis.
- Differential equations: Solving equations involving derivatives.
Example
Let's find the definite integral of secx from 0 to π/4:
∫[0,π/4] secx dx = ln|sec(π/4) + tan(π/4)| - ln|sec(0) + tan(0)|Simplifying, we get:
ln(√2 + 1) - ln(1) = ln(√2 + 1)Therefore, the definite integral of secx from 0 to π/4 is ln(√2 + 1).
