∫tanxdx
To evaluate the integral ∫tan(x)dx, we can use a trigonometric identity and then apply a technique called integration by substitution
To evaluate the integral ∫tan(x)dx, we can use a trigonometric identity and then apply a technique called integration by substitution.
The trigonometric identity we will use is:
sec^2(x) = 1 + tan^2(x)
Rearranging this identity, we get:
1 = sec^2(x) – tan^2(x)
Now, we can rewrite the integral as:
∫tan(x)dx = ∫(sec^2(x) – 1)dx = ∫sec^2(x)dx – ∫dx
The integral of sec^2(x) is a well-known trigonometric integral, which is:
∫sec^2(x)dx = tan(x) + C1, where C1 is the constant of integration.
The integral of dx is simply x + C2, where C2 is another constant of integration.
Therefore, the original integral becomes:
∫tan(x)dx = tan(x) + C1 – x + C2
Simplifying further, we have:
∫tan(x)dx = tan(x) – x + C, where C = C1 + C2 is the constant of integration.
So, the final answer to the integral is:
∫tan(x)dx = tan(x) – x + C, where C is a constant.
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