Trigonometric Integrals Trigonometric Integrals There are four - PowerPoint PPT Presentation

Trigonometric Integrals

Trigonometric Integrals There are four cases:

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x :

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x :

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x : � cos 2 xdx

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x : � cos 2 xdx 3. Integrals involving tan x and sec x :

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x : � cos 2 xdx 3. Integrals involving tan x and sec x : � tan 3 xdx

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x : � cos 2 xdx 3. Integrals involving tan x and sec x : � tan 3 xdx 4. Just tricky trig integrals (use any trig identity you know):

Trigonometric Integrals There are four cases: 1. Odd power of sin x or cos x : � sin 3 xdx 2. Even power of sin x or cos x : � cos 2 xdx 3. Integrals involving tan x and sec x : � tan 3 xdx 4. Just tricky trig integrals (use any trig identity you know): � sin 2 xdx cos 4 x + sin 4 x

Odd Powers of cos x and sin x � sin 3 xdx

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as:

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity:

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � sin 2 x sin xdx =

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � sin 2 x sin xdx =

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � � sin 2 x sin xdx = 1 − cos 2 x �

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � � sin 2 x sin xdx = 1 − cos 2 x � sin xdx

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � � sin 2 x sin xdx = 1 − cos 2 x � sin xdx � � sin x − cos 2 x sin x � = dx

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � � sin 2 x sin xdx = 1 − cos 2 x � sin xdx � � � sin x − cos 2 x sin x � = dx = sin xdx −

Odd Powers of cos x and sin x � sin 3 xdx We can write this integral as: � sin 2 x sin xdx Let’s use the identity: sin 2 x = 1 − cos 2 x � � � sin 2 x sin xdx = 1 − cos 2 x � sin xdx � � � � sin x − cos 2 x sin x cos 2 x sin xdx � = dx = sin xdx −

Odd Powers of cos x and sin x � � cos 2 x sin xdx = sin xdx −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � cos 2 x sin xdx = sin xdx −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � cos 2 x sin xdx = − cos x − sin xdx −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution:

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: u = cos x ,

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � cos 2 x sin xdx = − cos x −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � cos 2 x sin xdx = − cos x − − cos x −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � cos 2 x sin xdx = − cos x − u 2 − cos x −

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x − u 2 − cos x − dx

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x − u 2 − cos x − dx dx

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x − ✚ u 2 ✚ − cos x − dx ✚ ✚ dx

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx � = − cos x +

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx � u 2 du = − cos x +

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx � u 2 du = − cos x + = − cos x +

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx u 2 du = − cos x + u 3 � = − cos x + 3

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx u 2 du = − cos x + u 3 � = − cos x + 3 + C

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx u 2 du = − cos x + u 3 � = − cos x + 3 + C � sin 3 xdx =

Odd Powers of cos x and sin x ✯ − cos x ✟✟✟✟✟ � � � cos 2 x sin xdx = − cos x − cos 2 x sin xdx sin xdx − Now, we solve this integral by substitution: du u = cos x , dx = − sin x � � � � − du cos 2 x sin xdx = − cos x � ✚ u 2 ✚ − cos x − � � � dx ✚ − ✚ dx u 2 du = − cos x + u 3 � = − cos x + 3 + C � sin 3 xdx = − cos x +