What is the integral of sec^3(x)?

The integral of sec³(x) can be determined using integration techniques that involve trigonometric identities and substitution.

The integral is given by:

∫ sec³(x) dx = (1/2) sec(x) tan(x) + (1/2) ln |sec(x) + tan(x)| + C

Here’s a breakdown of how we arrive at this solution:

1. Recall the identity: sec²(x) = 1 + tan²(x). Using this, we can rewrite sec³(x) as sec(x) * sec²(x).
2. With that, we apply integration by parts. Let:
• u = sec(x) (which means du = sec(x) tan(x) dx)
• dv = sec²(x) dx (which means v = tan(x))
3. Applying integration by parts formula ∫ u dv = uv – ∫ v du, we get:

∫ sec³(x) dx = sec(x) tan(x) – ∫ tan(x) sec(x) tan(x) dx

4. The last integral simplifies to ∫ sec(x) tan²(x) dx, which can be rewritten as ∫ sec(x) (sec²(x) – 1) dx.
5. Breaking this into two separate integrals gives us:

∫ sec³(x) dx – ∫ sec(x) dx

6. Now we set up the equation:

∫ sec³(x) dx = sec(x) tan(x) – ∫ sec³(x) dx + ∫ sec(x) dx

7. Solving for ∫ sec³(x) dx and accumulating all terms leads us to the final result:

∫ sec³(x) dx = (1/2) sec(x) tan(x) + (1/2) ln |sec(x) + tan(x)| + C

And that’s how you compute the integral of sec³(x)!