∫ sec^2 x
To evaluate the integral of sec^2(x), we can use a technique called integration by substitution
To evaluate the integral of sec^2(x), we can use a technique called integration by substitution.
First, let’s recall a trigonometric identity: sec^2(x) = 1 + tan^2(x). Using this identity, we can rewrite the integral as:
∫ (1 + tan^2(x)) dx
Now, let’s make a substitution by letting u = tan(x). Taking the derivative of both sides with respect to x, we get du/dx = sec^2(x). Rearranging this equation, we obtain dx = du/sec^2(x).
Substituting u and dx into the integral, we have:
∫ (1 + tan^2(x)) dx = ∫ (1 + u^2) (du / sec^2(x))
Next, we need to express sec^2(x) in terms of u. We can use another trigonometric identity: sec^2(x) = 1 + tan^2(x). Therefore, sec^2(x) = 1 + u^2.
Substituting this into the integral, we get:
∫ (1 + u^2) (du / sec^2(x)) = ∫ (1 + u^2) du / (1 + u^2)
Now, notice that the (1 + u^2) terms cancel out, leaving us with:
∫ du
Finally, integrating du with respect to u gives us u + C, where C is the constant of integration.
Therefore, the final result is:
∫ sec^2(x) dx = tan(x) + C
In conclusion, the integral of sec^2(x) is equal to tan(x) plus a constant.
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