The integral of sec³(x) can be evaluated using integration techniques in calculus. It can be found by using a standard integration formula. Here’s how to compute it step by step:
Integral of Sec³(x)
To find the integral of sec³(x), we can use the following integral formula:
∫ sec³(x) dx = rac{1}{2} sec(x) tan(x) + rac{1}{2} ln | sec(x) + tan(x) | + C
Steps to Derive the Integral:
1. **Use Integration by Parts**: Sometimes, we can evaluate this integral by rewriting sec³(x) as sec(x) * sec²(x) and then applying the technique of integration by parts:
You let:
- u = sec(x) (the function we can differentiate easily)
- dv = sec²(x) dx (the function we can integrate easily)
2. **Differentiate and Integrate**: The next steps involve finding du and v:
- du = sec(x) tan(x) dx
- v = tan(x)
3. **Apply the Formula**: Now, apply the integration by parts formula:
∫ u dv = uv – ∫ v du
So, we substitute into the formula:
∫ sec³(x) dx = sec(x) tan(x) – ∫ tan(x) sec(x) tan(x) dx
4. **Simplify the Remaining Integral**: The integral that remains (∫ sec(x) tan²(x) dx) can be rewritten as:
Recall that tan²(x) = sec²(x) – 1, thus:
∫ sec(x) tan²(x) dx = ∫ sec³(x) dx – ∫ sec(x) dx
5. **Rearranging and Solving**: By bringing this all together, we can isolate the integral:
∫ sec³(x) dx = sec(x) tan(x) – ∫ sec³(x) dx + ∫ sec(x) dx
When you solve for ∫ sec³(x) dx, you will have:
2 ∫ sec³(x) dx = sec(x) tan(x) + ∫ sec(x) dx
Solving these integrals gives you:
∫ sec(x) dx = ln | sec(x) + tan(x) | + C
6. **Final Expression**: Thus, putting everything together leads us to the final answer:
∫ sec³(x) dx = rac{1}{2} sec(x) tan(x) + rac{1}{2} ln | sec(x) + tan(x) | + C
Conclusion
So, the integral of sec³(x) is not just a tedious computation, but rather an engaging puzzle. By using integration by parts and some clever substitutions, we arrive at a beautifully structured solution. Remember that practice is key in mastering these integrals, so don’t hesitate to try it out with different angles, and happy calculating!