Mastering the Integral of sec^2(x) | Techniques for Integration by Substitution

∫ 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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