# integral of secant^3 x

The integral of secant^3 x is (1/2){sec(x)tan(x) + ln|sec(x) + tan(x)|} + C

## The technique of the integral of secant^3 x dx

To find the integral of sec^3(x), we can use a technique called integration by parts. Here's the step-by-step solution:

Let's start by writing the integral:

∫sec^3(x) dx

To apply integration by parts, we choose u and dv as follows:

u = sec(x) (taking derivative, du = sec(x) * tan(x) dx)

dv = sec^2(x) dx

Now, let's find du and v:

du = sec(x) * tan(x) dx

v = ∫sec^2(x) dx = tan(x)

Using the integration by parts formula, which states ∫u dv = uv - ∫v du, we can rewrite the integral:

∫sec^3(x) dx

= ∫u dv

= uv - ∫v du

= sec(x) * tan(x) - ∫tan(x) * sec(x) * tan(x) dx

= sec(x) * tan(x) - ∫tan^2(x) * sec(x) dx

At this point, we can simplify further using the identity 1 + tan²x = sec²x. Rearranging this identity, we get

tan²(x) = sec²(x) - 1. Substituting this into the integral:

∫sec³x dx

= secx tanx - ∫(sec²x - 1) sec(x) dx

= sec(x) * tan(x) - ∫sec³(x) dx + ∫sec(x) dx

Now, let's isolate the integral on the left side:

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

Solving for the integral, we divide both sides by 2:

∫sec³(x) dx = (1/2) * (sec(x) * tan(x) + ∫sec(x) dx)

The integral of sec(x) can be found using a logarithmic identity:

∫sec(x) dx = ln|sec(x) + tan(x)| + C

Finally, we can substitute this result back into the equation:

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

Therefore, the integral of sec^3(x) is given by (1/2){sec(x)tan(x) + ln|sec(x) + tan(x)|} + C. where C is the constant of integration.

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