2 - 7 further calculus

Question 1

differentiateing inverse trig

Answer 1

y = sin^-1 x find dy/dx
siny = x
dx/dy = cosh y
dy/dx = 1/cos y
= 1/root(1-sin^2y)
= 1/root(1-x^2)

Question 2

differentiating arccosh(x/2)

Answer 2

use the definition of d/dx accosh(x)
but replace x with x/2 and times by the derivative of x/2 = 1/2

Question 3

using derivatives of inverse trig to get integration results

Answer 3

all in formula booklet except cos^-1 which is just -sin^-1
and tanh^-1 = ∫1/1-x^2

Question 4

partial fractions with a quadratic denominator

Answer 4

• factprise to linear factors
• perfect square also factorises
• doesnt factorise - make numerator Bx + C

Question 5

how to prove the integrals eg ∫ 1/√1-x² = arcsin x

Answer 5

x = sin u
dx/du = cos u
dx = cos u du
∫ 1/√1-sin²u *cosu du = ∫1/cosu * cosu du
= u
= arcsin(x) + c

Question 6

integrating where the denominator is not in the from x^2 + a^2 or √x^2 + a^2

Answer 6

complete the square to get it in this form
eg ∫ 1/(4x²- 12x -7)
4x²- 12x -7 = (2x -3)² - 16