integral of cos 6x
Understand the Problem
The question is asking for the integral of the cosine function multiplied by 6 times the variable x. We will solve it by applying the integration rules for trigonometric functions.
Answer
$$ \int 6x \cos(x) \, dx = 6x \sin(x) + 6 \cos(x) + C $$
Answer for screen readers
$$ \int 6x \cos(x) , dx = 6x \sin(x) + 6 \cos(x) + C $$
Steps to Solve
- Identify the integral to solve
We need to solve the integral of the function (6x \cos(x)). This can be written as:
$$ \int 6x \cos(x) , dx $$
- Apply integration by parts
To solve this integral, we will use integration by parts, which is based on the formula:
$$ \int u , dv = uv - \int v , du $$
Here, we will choose:
- ( u = 6x ) which implies ( du = 6 , dx )
- ( dv = \cos(x) , dx ) which implies ( v = \sin(x) )
- Substitute and calculate
Substituting these into the integration by parts formula gives us:
$$ \int 6x \cos(x) , dx = 6x \sin(x) - \int 6 \sin(x) , dx $$
- Integrate the remaining integral
Now we need to calculate the remaining integral:
$$ \int 6 \sin(x) , dx $$
The integral of (\sin(x)) is (-\cos(x)), so we have:
$$ \int 6 \sin(x) , dx = -6 \cos(x) $$
- Combine results
Now substituting back, we have:
$$ \int 6x \cos(x) , dx = 6x \sin(x) + 6 \cos(x) + C $$
where (C) is the constant of integration.
$$ \int 6x \cos(x) , dx = 6x \sin(x) + 6 \cos(x) + C $$
More Information
This result shows that by using integration by parts, we can effectively break down the integral involving the product of a polynomial and a trigonometric function. It's often used in calculus to handle more complex integrals.
Tips
- Forgetting to include the constant of integration (C). Always remember to add this after your integral calculation.
- Misidentifying (u) and (dv). Choosing the right functions for integration by parts is crucial for the simplification of the integral.
