{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 8.4 - 8.4 1 8.4 2 8.4 Trigonometric Integrals Products of...

This preview shows pages 1–3. Sign up to view the full content.

8.4 1 8.4 Trigonometric Integrals Products of Powers of Sines and Cosines We wish to evaluate integrals of the form: integraldisplay sin m x cos n xdx where m and n are nonnegative integers. Recall the double angle formulas for the sine and cosine functions. sin2 x = 2 sin x cos x cos 2 x = cos 2 x - sin 2 x The second formula can be used to to derive the very important “trig reduction” formulas. cos 2 x = 1 2 (1 + cos 2 x ) (1) sin 2 x = 1 2 (1 - cos 2 x ) (2) 8.4 2 These formulas are very useful when integrating even powers of sine and cosine. Example 1. Evaluate the following integrals. a. integraldisplay cos 2 xdx By (1) we have integraldisplay cos 2 xdx = 1 2 integraldisplay (1 + cos 2 x ) dx = 1 2 parenleftbigg x + sin2 x 2 parenrightbigg + C b. integraldisplay sin 4 xdx Using reduction (twice) we have sin 4 x = ( sin 2 x ) 2 = 1 4 (1 - cos2 x ) 2 = 1 4 ( 1 - 2cos2 x + cos 2 2 x ) = 1 4 parenleftbigg 1 - 2 cos2 x + 1 2 (1 + cos4 x ) parenrightbigg = 1 8 (3 - 4cos 2 x + cos 4 x )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8.4 3 It follows that integraldisplay sin 4 xdx = 1 8 integraldisplay (3 - 4cos 2 x + cos 4 x ) dx = 1 8
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

8.4 - 8.4 1 8.4 2 8.4 Trigonometric Integrals Products of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online