Integration of tan x
Understand the Problem
The question is asking for the integral of the function tan x. This involves finding the antiderivative of tan x, which is a process that requires an understanding of integral calculus.
Answer
$-\ln|\cos x| + C$
Answer for screen readers
The integral of tan x is -ln|cos x| + C, where C is the constant of integration.
Steps to Solve
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Recall integration by parts
We can use the integration by parts formula, but a more direct method is to use a trigonometric identity. Recall that $\tan x = \frac{\sin x}{\cos x}$.
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Rewrite the integral
Rewrite the integral using this identity:
$$\int \tan x dx = \int \frac{\sin x}{\cos x} dx$$
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Substitute u = cos x
Make the substitution $u = \cos x$. Then $du = -\sin x dx$, or $-du = \sin x dx$.
This transforms our integral to:
$$-\int \frac{1}{u} du$$
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Integrate
The integral of $\frac{1}{u}$ is $\ln|u|$. So we have:
$$-\int \frac{1}{u} du = -\ln|u| + C$$
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Substitute back
Now substitute back $u = \cos x$:
$$-\ln|u| + C = -\ln|\cos x| + C$$
This is our final answer.
The integral of tan x is -ln|cos x| + C, where C is the constant of integration.
More Information
The natural logarithm in the answer arises from the integration of 1/x. The absolute value is necessary because the cosine function can be negative, and we can't take the logarithm of a negative number. The constant C represents the fact that differentiation eliminates constants, so when we integrate, we need to account for all possible constants.
Tips
A common mistake is to forget the negative sign or the absolute value bars in the answer. Remember that when you substitute u = cos x, you get a negative sign from du = -sin x dx. Also, don't forget to include the constant of integration C.
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