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nameVoid
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Homework Statement
Homework Statement
The Attempt at a Solution
u= cos(2x) = > du= -2 sin(2x)
dv=cosx(2x) =>v= 1/2 sin(2x)
?
And the formula for integration by parts isnameVoid said:Homework Statement
Homework Statement
The Attempt at a Solution
u= cos(2x) = > du= -2 sin(2x)
dv=cosx(2x) =>v= 1/2 sin(2x)
?
Integration by Parts is a method used in calculus to solve integrals that involve the product of two functions. It involves using a formula that breaks down the integral into simpler parts, making it easier to solve. The formula is: ∫ u dv = uv - ∫ v du, where u and v are functions of x.
In general, it is best to choose u as the function that becomes simpler when differentiated, and v as the function that becomes easier to integrate when differentiated. In the case of cos(2x) and cosx(2x), u = cosx and dv = cos(2x) dx.
Step 1: Identify u and dv. In this case, u = cosx and dv = cos(2x) dx.Step 2: Differentiate u to find du. In this case, du = -sinx dx.Step 3: Integrate dv to find v. In this case, v = (1/2)sin(2x) + C.Step 4: Plug in u, v, du, and dv into the integration by parts formula: ∫ u dv = uv - ∫ v du.Step 5: Simplify the integral on the right side of the equation until you can solve for the original integral.
Yes, there are a few special cases to keep in mind. One is when the integral goes to infinity, in which case you may need to use a different method. Another is when the integral involves a logarithmic function, in which case you may need to use integration by parts multiple times.
To check if you have solved the integral correctly, you can differentiate the solution and see if it matches the original integrand. You can also use online integration calculators to verify your solution.