Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Double-Angle Formulas and Integration**
To solve the integral using double-angle formulas, we consider the expression given below, with the constant of integration denoted by C:
\[
\int \sin^2(x) \cos^2(2x) \, dx
\]
**Step-by-Step Solution:**
1. **Rewriting Using Trigonometric Identities**:
- Apply the identity for \(\cos^2(2x)\):
\[
\cos^2(2x) = \frac{1 + \cos(4x)}{2}
\]
- Similarly, for \(\sin^2(x)\), use:
\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]
2. **Substitute into the Integral**:
Replace the terms in the integral:
\[
\int \left(\frac{1 - \cos(2x)}{2}\right) \left(\frac{1 + \cos(4x)}{2}\right) \, dx
\]
3. **Simplification**:
Distribute and simplify the integrand:
\[
\int \frac{(1 - \cos(2x))(1 + \cos(4x))}{4} \, dx
\]
4. **Integration**:
Perform integration using standard trigonometric formulas and substitution if necessary.
5. **Conclusion**:
Evaluate the integral to obtain the result, adding the constant of integration, C.
By recognizing and applying trigonometric identities, particularly the double-angle formulas, we can simplify and solve integrals involving trigonometric expressions like the one above efficiently.
**Note**: Details of the final integration steps would typically follow these simplifications, involving expanded trigonometric terms.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd156a87-a341-4386-9d0b-4db0ff9d02ea%2F0f6d67c1-bc3d-4d4e-840d-107f2b3bf9f5%2Fotgsjw_processed.png&w=3840&q=75)
Transcribed Image Text:**Double-Angle Formulas and Integration**
To solve the integral using double-angle formulas, we consider the expression given below, with the constant of integration denoted by C:
\[
\int \sin^2(x) \cos^2(2x) \, dx
\]
**Step-by-Step Solution:**
1. **Rewriting Using Trigonometric Identities**:
- Apply the identity for \(\cos^2(2x)\):
\[
\cos^2(2x) = \frac{1 + \cos(4x)}{2}
\]
- Similarly, for \(\sin^2(x)\), use:
\[
\sin^2(x) = \frac{1 - \cos(2x)}{2}
\]
2. **Substitute into the Integral**:
Replace the terms in the integral:
\[
\int \left(\frac{1 - \cos(2x)}{2}\right) \left(\frac{1 + \cos(4x)}{2}\right) \, dx
\]
3. **Simplification**:
Distribute and simplify the integrand:
\[
\int \frac{(1 - \cos(2x))(1 + \cos(4x))}{4} \, dx
\]
4. **Integration**:
Perform integration using standard trigonometric formulas and substitution if necessary.
5. **Conclusion**:
Evaluate the integral to obtain the result, adding the constant of integration, C.
By recognizing and applying trigonometric identities, particularly the double-angle formulas, we can simplify and solve integrals involving trigonometric expressions like the one above efficiently.
**Note**: Details of the final integration steps would typically follow these simplifications, involving expanded trigonometric terms.
![**Evaluate the Definite Integral**
Calculate the integral below:
\[
\int_{0}^{\pi/2} \sqrt{1 - \cos(2x)} \, dx
\]
(A solution box is provided for the answer entry.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdd156a87-a341-4386-9d0b-4db0ff9d02ea%2F0f6d67c1-bc3d-4d4e-840d-107f2b3bf9f5%2Fkghi7ts_processed.png&w=3840&q=75)
Transcribed Image Text:**Evaluate the Definite Integral**
Calculate the integral below:
\[
\int_{0}^{\pi/2} \sqrt{1 - \cos(2x)} \, dx
\]
(A solution box is provided for the answer entry.)
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