Inverse Trigonometry Worksheet

Integration with Mathematica

Although I want you to be able to compute many types of integrals by hand, it is useful to be able to check your answers with some kind of technology like Mathematica.

Indefinite integrals

To compute the indefinite integral ∫(((x^3)+1)/((x^2)+x+1))d x using Mathematica, enter

Integrate[(x^3+1)/(x^2+x+1),x]

Definite integrals

To compute the definite integral (∫_4^7)((5x+1)/((x^3)(Sqrt[(x^2)-9])))d x using Mathematica, enter

Integrate[(5x+1)/(x^3*Sqrt[x^2-9]),{x,4,7}]

Review

Inverse sine

y=(sin^(-1))(x) ⟷ sin(y)=x

Domain is [-1,1], range is [-π/2,π/2].

(d/(d x))((sin^(-1))(x))=(1/(Sqrt[1-(x^2)]))

Inverse cosine

y=(cos^(-1))(x) ⟷ cos(y)=x

Domain is [-1,1], range is [0,π].

(d/(d x))((cos^(-1))(x))=-(1/(Sqrt[1-(x^2)]))

Inverse tangent

y=(tan^(-1))(x) ⟷ tan(y)=x

Domain is (-∞,∞), range is (-π/2,π/2).

(d/(d x))((tan^(-1))(x))=(1/((x^2)+1))

Inverse secant

y=(sec^(-1))(x) ⟷ sec(y)=x

Domain is (-∞,-1]⋃[1,∞), range is not standard—either [0,π/2)⋃(π/2,π] or [0,π/2)⋃[π,3π/2) .

(d/(d x))((sec^(-1))(x))=(1/(x(Sqrt[(x^2)-1]))) if x>1.

Problems to work and submit

From above we know ∫(1/((x^2)+1))d x = (tan^(-1))(x)+C. What about ∫(1/((x^2)+4))d x ? Start by rewriting this as ∫(1/(4((x^2)/4+1)))d x.

What is an appropriate substitution to make: u=?

Use that substitution to evaluate ∫(1/((x^2)+4))d x.

More generally, evaluate ∫(1/((x^2)+(a^2)))d x where a is a positive number.

Let h(x)=(sin^(-1))(2x-1).

What is its domain?

Find an equation for its tangent line at x=1/4.

Consider the function f(x)=(tan^(-1))(2x)-x.

What is the domain of f(x)? Write it in the form (A,B).

Compute (lim_(x->A))f(x) and (lim_(x->B))f(x).

Find any critical numbers for f(x). For each one determine if it is a local maximum, a local minimum or neither.

Based on the information you have computed, give a rough sketch of y=f(x).

Consider the function g(x)=(1/(x(Sqrt[(x^2)-25]))).

What is the domain of g(x)?

Find the area of the region bounded by y=g(x), the x-axis, and the two vertical lines x=6 and x=9.