Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Calculus

7. Antiderivatives & Indefinite Integrals

Integrals of Trig Functions

Previous problemNext problem

1:04 minutes

Problem 8.3.1

Textbook Question

1. State the half-angle identities used to integrate sin²x and cos²x.

Verified step by step guidance

The half-angle identities are trigonometric formulas that simplify expressions involving squared sine and cosine functions. They are derived from the double-angle identities.

The half-angle identity for sin²x is:

\sin^{2} x = \frac{1 - \cos 2 x}{2}

. This identity expresses sin²x in terms of cos(2x), making integration easier.

The half-angle identity for cos²x is:

\cos^{2} x = \frac{1 + \cos 2 x}{2}

. This identity expresses cos²x in terms of cos(2x), simplifying integration.

To integrate sin²x or cos²x, substitute the respective half-angle identity into the integral. For example, replace sin²x with

\frac{1 - \cos 2 x}{2}

After substitution, simplify the integral and proceed with standard integration techniques, such as integrating cos(2x) using the formula

\frac{\sin}{2 x}

divided by 2.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Half-Angle Identities

Half-angle identities are trigonometric identities that express the sine and cosine of half an angle in terms of the sine and cosine of the full angle. They are derived from the double angle formulas and are particularly useful in integration and simplifying trigonometric expressions. The identities are: sin²(x/2) = (1 - cos(x))/2 and cos²(x/2) = (1 + cos(x))/2.

Integration of Trigonometric Functions

Integrating trigonometric functions often requires the use of identities to simplify the integrand. For sin²x and cos²x, applying the half-angle identities allows us to rewrite these functions in a more manageable form, facilitating the integration process. This technique is essential for solving integrals that involve squared trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. They are fundamental tools in calculus for simplifying expressions and solving equations. Understanding these identities, including Pythagorean, angle sum, and half-angle identities, is crucial for effectively manipulating and integrating trigonometric functions.