A Guide on Finding the Integral of Sec(x) Using Integration by Substitution

∫secxdx

To find the integral of sec(x) with respect to x, we can use the technique of integration by substitution

To find the integral of sec(x) with respect to x, we can use the technique of integration by substitution.

Let’s start by recalling the derivative of the secant function, which is d/dx(sec(x)) = sec(x) * tan(x). We can rewrite this as:

sec(x) * tan(x) = d/dx(sec(x))

Now, let’s rewrite the integral using this relation:

∫sec(x)dx = ∫(sec(x) * tan(x))dx

Now, let’s substitute u = sec(x), so du = sec(x) * tan(x) dx. Rewriting the integral using the new variable gives us:

∫sec(x)dx = ∫(u * du), since sec(x) * tan(x) dx = du

The new integral becomes:

∫u du

Integrating ∫u du is straightforward, as it follows the power rule for integration. The power rule states that the integral of u^n with respect to u is (u^(n+1))/(n+1).

Applying the power rule to our integral, we get:

∫u du = (u^2)/2 + C

Substituting back u = sec(x), we have:

∫sec(x)dx = (sec^2(x))/2 + C

Therefore, the integral of sec(x) with respect to x is equal to (sec^2(x))/2 + C, where C is the constant of integration.

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