Math Problem Statement
Вычислить определенный интеграл
Solution
The given integral is:
To compute this definite integral, let's proceed with the following steps:
Step 1: Substitution
We'll start by substituting . Thus, or equivalently .
When , .
When , .
Substituting into the integral, we get:
Step 2: Simplifying the Integral
The integral becomes:
Since the limits of integration are reversed, we can switch the bounds and remove the negative sign:
Step 3: Integration by Parts
Now, apply integration by parts. Let:
- so that
- so that
The integration by parts formula is:
Substituting into the formula:
Step 4: Simplifying the Second Integral
Notice that:
So the second integral simplifies to:
This can be split into two integrals:
The first integral is straightforward:
The second integral is the arctangent function again:
Final Answer
Putting everything together, the definite integral evaluates to:
Simplify further to get the final numerical result:
This is the exact value of the integral.
Do you need any further details or have any questions?
Here are 8 related questions you might find interesting:
- How do you perform integration by parts in general?
- What are the properties of the arctan function?
- How does substitution simplify integrals?
- Why is the integral of equal to arctan(u)?
- How can definite integrals be interpreted geometrically?
- What are some common strategies for solving definite integrals?
- How do you handle integrals with inverse trigonometric functions?
- What other methods can be used to evaluate difficult integrals?
Tip: When evaluating definite integrals involving inverse trigonometric functions, consider using trigonometric identities or substitutions to simplify the expression.
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Math Problem Analysis
Mathematical Concepts
Definite Integral
Integration Techniques
Substitution Method
Integration by Parts
Inverse Trigonometric Functions
Formulas
Integration by Parts formula
Substitution method for integrals
Theorems
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Suitable Grade Level
College Level