Explanation Given: • ∫ sin 4 x ⋅ cos 2 x ⋅ d x Key Concepts Used: Many trigonometric functions can be integrated by combining substitution and Integration by parts with the appropriate trigonometric identities . Here in the given problem we have to find ∫ sin 4 x ⋅ cos 2 x ⋅ d x .that means we have to find the integral value of sin 4 x ⋅ cos 2 x . As per the method done in example sums, we have to split off a factor. We have to find the final value of integral using the reduction formula and repeatedly here we use that formula for three times to obtain the value of the given integral. Formula Used: Important Trigonometric Identities: Description: Identity Identity 1 :Pythagorean Identity sin 2 x + cos 2 x = 1 Identity 2 :Half-angle Identity for sin 2 x sin 2 x = 1 − cos 2 x 2 Identity 3 :Half-angle Identity for cos 2 x cos 2 x = 1 + cos 2 x 2 Identity 4 :Product Identity sin x ⋅ cos x = 1 2 sin 2 x Calculation: Description: Step 1 : Write the given integral. ∫ sin 4 x ⋅ cos 2 x ⋅ d x Step 2 : By using identity cos 2 x = 1 − sin 2 x simplify the given trigonometric function. sin 4 x ⋅ cos 2 x = sin 4 x ⋅ ( 1 − sin 2 x ) = sin 4 x − sin 6 x Step 3 : By using reduction Formula for Sine, sin n x for any exponent n , even or odd can be calculated. Here we have to apply that formula to find sin 6 x . ∫ sin n x ⋅ d x = − 1 n sin n − 1 x ⋅ cos x + n − 1 n ∫ sin n − 2 x ⋅ d x ∫ sin 6 x ⋅ d x = − 1 6 sin 6 − 1 x ⋅ cos x + 6 − 1 6 ∫ sin 6 − 2 x ⋅ d x ∫ sin 6 x ⋅ d x = − 1 6 sin 5 x ⋅ cos x + 5 6 ∫ sin 4 x ⋅ d x Step 4 : Substitute the above obtained value in given integration. ∫ sin 4 x ⋅ cos 2 x ⋅ d x = ∫ ( sin 4 x − sin 6 x ) ⋅ d x = ∫ sin 4 x ⋅ d x − ∫ sin 6 x ⋅ d x = ∫ sin 4 x ⋅ d x − ( − 1 6 sin 5 x ⋅ cos x + 5 6 ∫ sin 4 x ⋅ d x ) = ∫ sin 4 x ⋅ d x + 1 6 sin 5 x ⋅ cos x − 5 6 ∫ sin 4 x ⋅ d x ∫ sin 4 x . cos 2 x ⋅ d x = 1 6 sin 5 x ⋅ cos x + 1 6 ∫ sin 4 x ⋅ d x → e q u a t i o n ( 1 ) Step 5 : By using reduction Formula for Sine, sin n x for any exponent n , even or odd can be calculated. Here we have to apply that formula again to find sin 4 x ...

3rd Edition

ISBN: 9781464114885

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

See similar textbooks

Question

Chapter 7.2, Problem 11E

To determine

The value of the given integral by using the methods employed in Example $4 a n d 5$

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 7.2, Problem 10E

Chapter 7.2, Problem 12E

Students have asked these similar questions

3. In the space below, describe in what ways the function f(x) = -2√x - 3 has been transformed from the basic function √x. The graph f(x) on the coordinate plane at right. (4 points) -4 -&- -3 -- -2 4 3- 2 1- 1 0 1 2 -N -1- -2- -3- -4- 3 ++ 4

2. Suppose the graph below left is the function f(x). In the space below, describe what transformations are occuring in the transformed function 3ƒ(-2x) + 1. The graph it on the coordinate plane below right. (4 points)

1 1. Suppose we have the function f(x) = = and then we transform it by moving it four units to the right and six units down, reflecting it horizontally, and stretching vertically by 5 units. What will the formula of our new function g(x) be? (2 points) g(x) =

Chapter 7 Solutions

Calculus: Early Transcendentals

Ch. 7.1 - Prob. 1PQ Ch. 7.1 - Prob. 2PQ Ch. 7.1 - Prob. 3PQ Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E

Knowledge Booster

The value of the given integral by using the methods employed in Example 4 a n d 5

The value of the given integral by using the methods employed in Example 4 and 5

Want to see the full answer?

Chapter 7 Solutions

The value of the given integral by using the methods employed in Example $4 a n d 5$