integration of tanx dx
Understand the Problem
The question is asking for the indefinite integral of the function tan(x) with respect to x. To solve this, we will use the fact that tan(x) can be expressed as sin(x)/cos(x) and apply the integration techniques for trigonometric functions.
Answer
The indefinite integral of \( \tan(x) \, dx \) is: $$ -\ln|\cos(x)| + C $$
Answer for screen readers
The indefinite integral of ( \tan(x) , dx ) is: $$ -\ln|\cos(x)| + C $$
Steps to Solve
- Rewrite the tangent function
Start by expressing the tangent function in terms of sine and cosine: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
- Use substitution method
Let ( u = \cos(x) ). Then, the derivative ( du ) is: $$ du = -\sin(x) , dx $$ or equivalently: $$ \sin(x) , dx = -du $$
- Substitute into the integral
Now substitute ( u ) and ( du ) into the integral: $$ \int \tan(x) , dx = \int \frac{\sin(x)}{\cos(x)} , dx = \int \frac{-du}{u} $$
- Integrate
Now integrate: $$ \int \frac{-du}{u} = -\ln|u| + C $$ where ( C ) is the constant of integration.
- Back substitute
Finally, replace ( u ) with ( \cos(x) ) to express the answer in terms of ( x ): $$ -\ln|\cos(x)| + C $$
The indefinite integral of ( \tan(x) , dx ) is: $$ -\ln|\cos(x)| + C $$
More Information
The integral of ( \tan(x) ) showcases the relationship between trigonometric functions and logarithms. The constant of integration ( C ) represents the infinite family of antiderivatives, since any constant added to a function has the same derivative.
Tips
- Forgetting to include the absolute value in ( \ln|\cos(x)| ), which is crucial for the domain of the logarithm.
- Not applying the negative sign when substituting the expression for ( \sin(x) , dx ).
