∫(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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