Math Problem Statement
solve the integral of 19(sinx)(cosx)dx using integration by parts
Solution
To solve the integral using integration by parts, we first need to apply a suitable strategy. However, this integral can be simplified before using integration by parts by recognizing the trigonometric identity:
So, we can rewrite the integral as:
Now, let's solve this integral:
Step 1: Simplify the Integral
Step 2: Integrate Using Substitution
Let , so or .
Substituting into the integral, we get:
Step 3: Integrate
The integral of is , so:
Step 4: Substitute Back
Substituting back into the equation, we get:
So, the integral is:
Would you like a more detailed explanation or have any questions about the process?
Here are five related questions you might consider:
- How would you solve the integral of without using substitution?
- What are some other trigonometric identities that can simplify integrals?
- How does integration by parts work in cases where substitution is not applicable?
- Can you derive the identity from basic trigonometric principles?
- What is the significance of the constant in indefinite integrals?
Tip: Always check if a trigonometric identity can simplify the integral before choosing a method like integration by parts!
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Math Problem Analysis
Mathematical Concepts
Calculus
Trigonometric Integrals
Integration by Substitution
Formulas
sin(2x) = 2sin(x)cos(x)
∫sin(u)du = -cos(u)
Theorems
Trigonometric Identity
Substitution Method for Integration
Suitable Grade Level
College Calculus or Advanced High School
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