How to Integrate tanx using Substitution Method | Antiderivative of tanx

∫tanx dx

ln|secx| + C

To integrate the function ∫tanx dx, we can use the substitution method.

Let u = cos(x) and du/dx = -sin(x)

Using trigonometric identity, we know that:

tan(x) = sin(x) / cos(x)

So, we can write:

∫tanx dx = ∫sin(x) / cos(x) dx

Now, substitute u = cos(x) into the integral:

∫sin(x) / cos(x) dx = ∫(1/u) * (-du)

We can simplify this expression:

∫(1/u) * (-du) = -∫(1/u) du

Using power rule of integration:

-∫(1/u) du = -ln|u| + C

Substituting back u = cos(x):

-∫(1/u) du = -ln|cos(x)| + C

So, the final answer is:

∫tanx dx = -ln|cos(x)| + C

Therefore, the antiderivative of tanx is -ln|cos(x)| + C.

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