Solving the Integral of sec(x)tan(x) using u-Substitution

∫ sec(x)tan(x) dx

To solve the integral of sec(x)tan(x), we can use a u-substitution

To solve the integral of sec(x)tan(x), we can use a u-substitution. Let’s rewrite the integral as:

∫ sec(x)tan(x) dx

We will let u = sec(x). Now, differentiating u with respect to x, we get:

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

Notice that du/dx looks very similar to the integrand. So, we can substitute u and du into the integral:

∫ sec(x)tan(x) dx = ∫ du

Since u = sec(x), we need to convert everything into terms of u. We can use the identity:

sec^2(x) = 1 + tan^2(x)

Rearranging, we have:

tan^2(x) = sec^2(x) – 1

Dividing both sides by sec^2(x), we obtain:

tan^2(x) / sec^2(x) = sec^2(x) / sec^2(x) – 1 / sec^2(x)

tan^2(x) / sec^2(x) = 1 – 1 / sec^2(x)

tan^2(x) / sec^2(x) = 1 – cos^2(x)

Now, substituting into the integral:

∫ sec(x)tan(x) dx = ∫ du

The integral of du is simply u + C, where C is the constant of integration. Therefore, the final result is:

∫ sec(x)tan(x) dx = sec(x) + C

So, the solution to the integral of sec(x)tan(x) dx is sec(x) + C, where C is the constant of integration.

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