Math Problem Statement
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Solution
To solve the integral , we will use the following steps:
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Substitution: Let . Then , which implies .
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Changing the Limits: When :
When :
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Rewrite the Integral: Substitute and into the integral:
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Evaluate the Integral: The integral of is :
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Apply the Limits: Evaluate from to : [ \frac{1}{2} \left[ -\cos\left( \frac{\pi}{2} \right) + \cos(0) \right] = \frac{1}{2} \left[ -0 + 1 \right] = \frac{1}{2} \left[ 1 \right] = \frac{1}{2} ]
So, the value of the integral is .
Would you like more details or have any questions about this solution? Here are some related questions for further practice:
- Evaluate .
- Find the antiderivative of .
- Determine .
- Calculate .
- Solve .
- Evaluate .
- Find .
- Determine .
Tip: When dealing with trigonometric integrals, using substitution can simplify the process significantly. Always remember to change the limits of integration when you make a substitution.
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Math Problem Analysis
Mathematical Concepts
Calculus
Trigonometry
Formulas
Substitution method
Integral of sine function
Theorems
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Suitable Grade Level
Advanced High School and Above