Evaluate the definite integral in the given problem

In summary, the conversation discusses the use of integration by parts to solve a problem, specifically in relation to the highlighted part. The main concern is whether there is an alternative method, but it is concluded that integration by parts is the most straightforward way. The relevant equations mentioned are for integration by parts and the conversation also references an attached homework statement.
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chwala
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Homework Statement
See attached.
Relevant Equations
Integration by parts.
My interest is on the highlighted part only. Find the problem and solution here.

1651270870747.png
1651270902092.png


This is clear to me (easy )...i am seeking an alternative way of integrating this...or can we say that integration by parts is the most straightforward way?

The key on solving this using integration by parts is to note that;
##u=t, du=dt, dv=sin \frac {1}{2} t, v=-2 cos \frac {1}{2} t##
 
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  • #2
chwala said:
Homework Statement:: See attached.
Relevant Equations:: Integration by parts.

can we say that integration by parts is the most straightforward way?
I am not sure but afraid that we have no better way.
 
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Related to Evaluate the definite integral in the given problem

1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function within a specific interval.

2. How do you evaluate a definite integral?

To evaluate a definite integral, you need to first find the anti-derivative of the function. Then, plug in the upper and lower limits of the integral into the anti-derivative and subtract the results to find the final value.

3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration, while an indefinite integral does not. This means that a definite integral will give a specific numerical value, while an indefinite integral will give a general function with a constant of integration.

4. What is the purpose of evaluating a definite integral?

Evaluating a definite integral is useful in many fields of science, such as physics and engineering. It allows us to find the total value of a function, which can represent physical quantities like distance, velocity, and acceleration.

5. Are there any shortcuts or tricks for evaluating definite integrals?

Yes, there are several techniques for evaluating definite integrals, such as substitution, integration by parts, and trigonometric substitution. These techniques can help simplify the integral and make it easier to evaluate.

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