## ∫(tanx)dx

### To integrate ∫(tanx)dx, we can use a trigonometric identity and rewrite it in terms of sine and cosine

To integrate ∫(tanx)dx, we can use a trigonometric identity and rewrite it in terms of sine and cosine.

The trigonometric identity we’ll use is:

tanx = sinx/cosx

Now, let’s rewrite the integral:

∫(tanx)dx = ∫(sinx/cosx)dx

To solve this integral, we can substitute u = cosx or du = -sinx dx. Notice that the numerator sinx matches the du, which is convenient.

Making this substitution, we get:

∫(sinx/cosx)dx = ∫(-du/u)

Now, we have a much simpler integral. We can proceed to solve this using the natural logarithm function.

∫(-du/u) = -∫(1/u)du = -ln|u| + C

Finally, we substitute back u = cosx:

-∫(1/u)du = -ln|cosx| + C

Therefore, the solution to the integral ∫(tanx)dx is -ln|cosx| + C, where C is the constant of integration.

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