Fundamental Concepts and Techniques of Calculus
Hints to the Exercises: Integral Calculus
Last updated: 041012
1. Integration By Parts: (c) Use the ﬁrst Pythagorean identity and rewrite
(a) Use integration by parts (R 4:6c) and set u = x cos3 x = cos2 x cos x = (1 − sin2 x) cos x.
and v = ex .
Then apply (R 4:5b).
(c) Use integration by parts (R 4:6c) and set u = x
(e) You can use the trigonometric identity cos2 x =
and v = sin x 1
2 (1 + cos 2x) to reduce the power of cos in
(d) Integrate by parts (R 4:6c) twice such that the the integrand or apply the recursion formula
remaining integral is a multiple of the original (R 4:5q).
integral. Combine those integrals on the left side
(f) Use the trigonometric identities
of the equation.
(e) sin x cos x = 2 sin(2x)
(f) cos x = 2 1 + cos(2x)
(g) sin x = 2 1 − cos(2x)
(h) to reduce the number of factors.
(q) 3. Substitution:
(r) (a) √ the substitution (R 4:6b) given by y = 1 +
2. Trigonometric Integrals:
(a) Use the Second Pythagorean identity and (e)
tan3 x sec3 x = sec2 x tan2 x(sec x tan x) (g) Try the substitution (R 4:6b) given by y =
= sec2 (sec2 x − 1)(sec x tan x).
Then apply (R 4:5b). (i)
(n) 5. Improper Integrals:
(s) Reduce the integrand by the substitution (e)
(R 4:6b) given by y = ex .
(t) Complete the square and use trigonometric sub-
stitution (R 4:6(b)iii). 6. Applications of Integration:
(u) (a) Area Between Two Curves:
(y) Use trigonometric substitution (R 4:6(b)iii) and (b) Volume By Slicing:
set x = a tan θ. i.
(z) let u = −x ii.
4. Partial Fractions:
(b) (c) Volume By Shells:
(d) Arc Length:
(j) (e) Surface Area:
(o) (f) Mass From Density:
(s) (g) Center of Mass and Centroid:
iv. (e) Find the Length of the Parametric Arc:
8. Polar Coordinates:
ix. (a) Write the Equation in Polar Coordinates:
(h) Fluid Pressure: i.
(i) Work: (b) Write the Equation in Cartesian Coordinates:
iv. (c) Sketch the Polar Curves:
7. Parametric Curves: i.
(a) Express the Parametric Curve by an Equation in
x and y:
(d) Calculate the Area enclosed by the Polar Curve:
(b) Find a Parametrization x = (t), y = y(t),
t ∈ [0, 1] for iii.
iii. (e) Find the Slope of the Polar Curve:
(d) Find the slope of the given curve at the given iii.
point and give an equation of the tangent line: (f) Find the Length of the Polar Curve: