Integrals With Only Even Powers cos 2 xdx Integrals With Only Even - PowerPoint PPT Presentation

Integrals With Only Even Powers � cos 2 xdx

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity:

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that:

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2 And if we square both sides:

Integrals With Only Even Powers � cos 2 xdx Here we need to use the trig identity: � 1 + cos θ cos θ 2 = 2 If we make x = θ 2 , we have that: � 1 + cos 2 x cos x = 2 And if we square both sides: cos 2 x = 1 + cos 2 x 2

Integrals With Only Even Powers So, we replace in our integral:

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � cos 2 xdx =

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � � cos 2 xdx =

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � cos 2 xdx = 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � cos 2 xdx = dx 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � cos 2 xdx = dx = 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � =

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 = 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 = 2 dx +

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � = 2 dx +

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x = 2 dx + 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x = 2 dx + dx 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x dx = 1 = 2 dx + 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 = 2 dx + 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 = 2 dx + dx + 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � dx = 1 dx + 1 = 2 dx + 2 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + 2 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + cos 2 x 2 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 dx + 1 = 2 dx + cos 2 xdx 2 2 2

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x = x 2+

Integrals With Only Even Powers So, we replace in our integral: cos 2 x = 1 + cos 2 x 2 � 1 + cos 2 x � � 1 � � 2 + cos 2 x cos 2 xdx = dx = dx 2 2 � 1 � cos 2 x � � dx = 1 +1 = 2 dx + cos 2 xdx dx 2 2 2 � �� � x � = x 2 + 1 cos 2 xdx 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution:

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u =

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u = 2 x

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: u = 2 x ,

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx =

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 x 2+

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 x 2 + 1 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � x 2 + 1 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � x 2 + 1 cos u 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du cos u . 2 2 dx

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du cos u . dx 2 2 dx

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ = x 2+

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ 2 + 1 = x 4

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 4

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � 2 + 1 x du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos u 4

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � 2 + 1 x du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu 4

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu 4 � �� � sin u

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu = x 4 2

Integrals With Only Even Powers � x 2 + 1 cos 2 xdx 2 We solve this integral by substitution: du u = 2 x , dx = 2 � 1 � � x 2 + 1 du ✚ cos u . dx ✚ ✚ 2 2 dx ✚ � = x 2 + 1 cos udu = x 4 2