Understanding how to find the integral of sec(x)tan(x) using trigonometric identities and u-substitution

∫(secxtanx)dx

To find the integral of sec(x)tan(x) with respect to x, we need to make use of trigonometric identities

To find the integral of sec(x)tan(x) with respect to x, we need to make use of trigonometric identities.

Firstly, we’ll recall the identity: sec(x) = 1/cos(x).

Rewriting the integral, we have:

∫(sec(x)tan(x))dx = ∫(1/cos(x) * sin(x)/cos(x))dx

Next, we can use u-substitution to simplify the integral.

Let u = cos(x), then du = -sin(x)dx.

Rewriting the integral with u, we obtain:

∫(1/u * -du) = -∫(1/u)du

Now, we can use the power rule of integration, which states that ∫(1/x)dx = ln|x| + C.

Applying this rule to our integral, we have:

-∫(1/u)du = -ln|u| + C

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

-∫(1/u)du = -ln|cos(x)| + C

Therefore, the integral of sec(x)tan(x) with respect to x is -ln|cos(x)| + C, where C represents the constant of integration.

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