Understanding Trigonometric Substitution and U Substitution

x = 3 sin(theta)

x = 3 cot(theta)

x = 3 tan(theta)

x = 3 cos(theta)

sin^2(theta) + cos^2(theta) = 1

tan^2(theta) + 1 = sec^2(theta)

1 + cot^2(theta) = csc^2(theta)

sin(theta) = 1/csc(theta)

To express the solution in terms of theta

To express the solution in terms of x

To simplify the integral further

To find the derivative

cos(theta) = 1 / sin(theta)

cos(theta) = sin(theta) / tan(theta)

cos(theta) = sqrt(1 - sin^2(theta))

cos(theta) = 1 - sin(theta)

sine(theta) = x^2 / 3

sine(theta) = 3 / x

sine(theta) = 3x

sine(theta) = x / 3

u = sqrt(1 - x^2 / 9)

u = 1 - x^2 / 9

u = x^2 / 9

u = 1 / sqrt(1 - x^2 / 9)

1/2

2/3

3/2

5/2

Using algebraic manipulation

Using calculus techniques

Using trigonometric identities

Using the right triangle method

b

a

k

c

To simplify the expression

To solve a differential equation

To evaluate the indefinite integral

To find the derivative