Mastering Integration | Step-by-Step Guide to Finding the Integral of Tangent Function

∫(tanx)dx

To find the integral of the tangent function, we can use a technique known as integration by substitution

To find the integral of the tangent function, we can use a technique known as integration by substitution. Let’s go through the steps:

1. Start by recognizing the integral you have: ∫(tanx)dx.

2. Notice that the derivative of tanx is sec^2x. This suggests that we should choose u = tanx as our substitution.

3. Differentiate u with respect to x to find du/dx. In this case, du/dx = sec^2x.

4. Rearrange the differential equation to solve for dx. Dividing both sides of du/dx = sec^2x by sec^2x, we get dx = du/sec^2x.

5. Substitute u = tanx and dx = du/sec^2x in the original integral. This will transform the integral into a new expression with only u terms: ∫(tanx)dx = ∫(u)(du/sec^2x).

6. Simplify the expression using the fact that sec^2x = 1 + tan^2x. Applying this identity, we have ∫(u)(du/(1 + u^2)).

7. Now, we can integrate the simplified expression. Breaking it into two separate integrals, we have ∫(u/(1 + u^2))du.

8. Split the integral into partial fractions for ease of integration. Write (u/(1 + u^2)) as (A/(1 + u) + B/(1 – u)).

9. Find the values of A and B by multiplying through by the denominators and equating the coefficients of corresponding powers of u. After solving, you’ll find that A = 1/2 and B = -1/2.

10. Rewriting the integral after the partial fractions decomposition, we have ∫(1/2)(du/(1 + u)) – ∫(1/2)(du/(1 – u)).

11. Integrate both of these individual fractions, resulting in (1/2)ln|1 + u| – (1/2)ln|1 – u| + C, where C is the constant of integration.

12. Substitute back u = tanx, and simplify the expression further: (1/2)ln|1 + tanx| – (1/2)ln|1 – tanx| + C.

And there is your final answer for the integral of tanx: (1/2)ln|1 + tanx| – (1/2)ln|1 – tanx| + C.

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