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