Trigonometric Integration

MAT 229, Spring 2021

Week 4

Supporting materials

If you wish to get a different perspective on the notes below, try either of the following textbook sections.

Stewart’s Calculus
	Section 7.2: Trigonometric integrals

Boelkins/Austin/Schlicker’s Active Calculus
	Section 5.3: Integration by substitution

Review

We have three antidifferentiation techniques:

Recognize it immediately as the derivative of some function, for example
	∫(e^x)d x=(e^x)+C

Try substituting for some internal part (the chain rule backwards). Don’t forget the differential. For example, to evaluate ∫(x^2)cos((x^3))d x use the substitution u=(x^3) and d u=3(x^2)d x to replace the above integral with the simpler one,
	∫(1/3)cos(u)d u=(1/3)sin(u)+C=(1/3)sin((x^3))+C

Integration by parts (the product rule backwards). Here it is in two different forms:
	a. ∫u d v=u v-∫v d u
	b. ∫u(x) v'(x)d x=u (x)v(x)-∫v(x) u'(x)d x

Today we introduce trigonometric integrals, as a special class of integrals that seem to arise frequently, and that involve particular trig substitutions.

Questions

Choose an appropriate technique to evaluate the following integrals.

Find the area of the region bounded above by y=x (e^(-x)),  below by the x-axis, and the line x=5.

Plot[x E^-x,{x,-1,5}]

Find the volume of the surface of revolution generated by rotating about the x-axis the region bounded above by y=x sec((x^3)),  below by the x-axis, and the line x=1.

Plot[x Sec[(x^3)],{x,-Pi/2,Pi/2}]
Integrate[Pi x^2 Sec[(x^3)]^2,{x,0,1}]
N[%]

Trigonometric identities

Pythagorean identities

(cos^2)(θ)+(sin^2)(θ)=1

Questions

Divide this equation by (cos^2)(θ) to get another Pythagorean identity.
	(((cos^2)(θ))/((cos^2)(θ)))+(((sin^2)(θ))/((cos^2)(θ)))=(1/((cos^2)(θ)))
⟶ 1+(tan^2)(θ)=(sec^2)(θ)

Divide this equation by (sin^2)(θ) to get another Pythagorean identity.
	(((cos^2)(θ))/((sin^2)(θ)))+(((sin^2)(θ))/((sin^2)(θ)))=(1/((sin^2)(θ)))
⟶ (cot^2)(θ)+1=(csc^2)(θ)

Double angle formulas

There are two identities (other than the Pythagorean) that I think are worth memorizing: the sine and cosine sum formulas are
	sin(α+β)=sin(α)cos(β)+sin(β)cos(α)
cos(α+β)=cos(α)cos(β)-sin(α)sin(β)

You can get the difference formulas from the sum formulas, using the symmetry properties of sines and cosines (so you don’t need to memorize those -- but you do need to know those symmetry properties!):  
	sin(α-β)=sin(α)cos(β)-sin(β)cos(α)
cos(α-β)=cos(α)cos(β)+sin(α)sin(β).

Questions

What is sin(2θ)=sin(θ+θ)? (Video)

What is cos(2θ)=cos(θ+θ)? (Video)

Rewrite this using a Pythagorean trigonometric identity so that cos(2θ) is in terms of only cos(θ).

Rewrite this using a Pythagorean trigonometric identity so that cos(2θ) is in terms of only sin(θ). (Video)

Sine squared and cosine squared

The results above give us two trigonometric identities that we will use below.

(cos^2)(θ)=((1+cos(2θ))/2)

(sin^2)(θ)=((1-cos(2θ))/2)

Trigonometric integrals

Here are some techniques for evaluating integrals of the following forms.

∫(sin^m)(x)(cos^n)(x)d x

∫(tan^m)(x)(sec^n)(x)d x

∫(cot^m)(x)(csc^n)(x)d x

Questions/review:

Integration is just differentiation in reverse: what are the following derivatives?

(d/(d x))cos(x)

(d/(d x))sin(x)

(d/(d x))tan(x)

(d/(d x))sec(x)

D[Sec[x],x]

(d/(d x))cot(x)

D[Cot[x],x]

(d/(d x))csc(x)

D[Csc[x],x]

Questions

Evaluate ∫(cos^4)(x)sin(x)d x using the substitution u=cos(x).(Video)

Integrate[Cos[x]^4 Sin[x],x]

Evaluate ∫(sin^6)(x)(cos^3)(x)d x by writing this as ∫(sin^6)(x)(cos^2)(x)cos(x)d x=∫(sin^6)(x)(1-(sin^2)(x))cos(x)d x. (Video)

Integrate[Sin[x]^6 Cos[x]^3 ,x]
Integrate[(Sin[x]^6(1-Sin[x]^2))Cos[x],x]

Find the area under the curve y=(sin^3)(x)(cos^3)(x) for 0≤x≤π/2.  (Video)

Plot[Sin[x]^3 Cos[x]^3,{x,0,Pi/2}]

Evaluate ∫(cot^4)(x)(csc^2)(x)d x. (Video)

Integrate[Cot[x]^4 Csc[x]^2,x]

Evaluate ∫(tan^2)(x)(sec^4)(x)d x by first writing (sec^4)(x)=(sec^2)(x)(sec^2)(x), using a Pythagorean identity on one of the factors, and then making an appropriate substitution. (Video)

Integrate[Tan[x]^2 Sec[x]^4,x]

Questions

Use the trigonometric identity (cos^2)(θ)=((1+cos(2θ))/2) to evaluate
	∫(cos^2)(θ)d θ=∫((1+cos(2θ))/2)d θ. (Video)

Find the volume of the solid of revolution obtained by rotating about the x-axis the region above the x-axis and one arch of the sine curve y=sin(x). (Video)

Plot[Sin[x],{x,0,Pi}]

Homework

IMath problems on trigonometric integrals.