∫tan(x)dx
To solve the integral of tan(x), we can use a technique called integration by parts
To solve the integral of tan(x), we can use a technique called integration by parts. This is a method that allows us to simplify the integral by rewriting it in terms of other functions.
The integral we need to solve is ∫tan(x)dx.
Step 1: Identify the parts
In integration by parts, we choose two parts of the integrand and assign them as u for one part and dv for the other part, where dx represents the differential.
Here, we can choose u = tan(x) and dv = dx.
Step 2: Calculate du and v
We need to find the differentials du and v by differentiating u and integrating dv, respectively.
For u = tan(x), we find du by differentiating u with respect to x:
du = sec^2(x) dx
For dv = dx, we integrate dv with respect to x:
v = x
Step 3: Apply the integration by parts formula
The integration by parts formula states:
∫u*dv = uv – ∫v*du
Using this formula, we can rewrite the original integral as:
∫tan(x)dx = x*tan(x) – ∫x*sec^2(x) dx
Step 4: Simplify the integral
The new integral in the equation above, ∫x*sec^2(x) dx, is simpler than the original integral. However, we are not done yet. This new integral requires more steps.
To solve ∫x*sec^2(x) dx, we can once again apply integration by parts.
Let’s assign u = x and dv = sec^2(x) dx.
Differentiating u, we get du = dx.
Integrating dv, we get v = tan(x).
Using the integration by parts formula, we rewrite the integral as:
∫x*sec^2(x) dx = x*tan(x) – ∫tan(x) dx
Notice that the second term in the equation is the same as the original integral we want to solve. Let’s call it I:
I = ∫tan(x) dx
Now we can substitute this value back into the equation:
∫x*sec^2(x) dx = x*tan(x) – I
Therefore, we have:
∫tan(x)dx = x*tan(x) – ∫x*sec^2(x) dx
∫tan(x)dx = x*tan(x) – (x*tan(x) – ∫tan(x) dx)
Step 5: Simplify further
Simplifying the equation, we get:
∫tan(x)dx = x*tan(x) – x*tan(x) + ∫tan(x) dx
The -x*tan(x) and x*tan(x) terms cancel each other out, leaving us with:
∫tan(x)dx = ∫tan(x) dx
Step 6: Solve for the integral
Now, we have the equation:
∫tan(x)dx = ∫tan(x) dx
By subtracting ∫tan(x) dx from both sides, we get:
0 = 0
This means that the integral of tan(x) is equal to zero.
In conclusion:
∫tan(x)dx = 0
More Answers:
Approaching the Limit: The Continuity of sin(x) as x Goes to 0Mastering the Integration of sin(x): Unveiling the Antiderivative and Fundamental Rule of Calculus
The Integral of Cos(x): A Step-by-Step Guide for Finding the Antiderivative