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Evaluating Multiple Integrals
Published on November 22, 2024
Calculus
The integrals were evaluated using trigonometric substitution, partial fraction decomposition, and substitution. Integral c) was solved using trigonometric substitution, resulting in x/(√(1-x^2)) + C. Integral d) diverged to infinity. Integral e) was solved using partial fraction decomposition, leading to a solution involving natural logarithms and a constant term.
Question
evaluate the following integrals
Answer
Let’s evaluate the integrals one by one.
c)
This integral can be solved using trigonometric substitution. Let
The integral becomes:
The integral becomes:
This integral is a standard one:
Since
d)
Let
When
The integral becomes:
When
The integral becomes:
The integral diverges.
e)
Notice that the derivative of the denominator is
First, factor the denominator:
So we want to find
First, factor the denominator:
So we want to find
Multiplying both sides by
If
If
Now, comparing the coefficients of
If
Now, comparing the coefficients of
Thus, the integral becomes:
We can write this as: