# Hints to the Exercises Integral Calculus

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```					                                                     Review
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).
(b)
(d)
(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.
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(e)                                                                             sin x cos x =   2   sin(2x)
2       1
(f)                                                                                  cos x =    2   1 + cos(2x)
2       1
(g)                                                                                  sin x =    2   1 − cos(2x)
(h)                                                                     to reduce the number of factors.
(i)                                                             (g)
(j)                                                             (h)
(k)                                                               (i)
(l)                                                              (j)
(m)                                                               (k)
(n)                                                               (l)

(o)                                                             (m)
(n)
(p)
(q)                                                          3. Substitution:
(r)                                                              (a) √ the substitution (R 4:6b) given by y = 1 +
Try
(s)                                                                    x.

(t)                                                             (b)
(c)
2. Trigonometric Integrals:
(d)
(a) Use the Second Pythagorean identity and                      (e)
rewrite                                                      (f)
tan3 x sec3 x = sec2 x tan2 x(sec x tan x)                (g) Try the substitution (R 4:6b) given by y =
sin−1 (2x).
= sec2 (sec2 x − 1)(sec x tan x).
(h)
Then apply (R 4:5b).                                       (i)
(b)                                                               (j)

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(k)                                                           (w)
(l)                                                           (x)
(m)                                                             (y)
(n)                                                        5. Improper Integrals:
(o)
(a)
(p)
(b)
(q)
(c)
(r)
(d)
(s) Reduce the integrand by the substitution                   (e)
(R 4:6b) given by y = ex .
(f)
(t) Complete the square and use trigonometric sub-
stitution (R 4:6(b)iii).                              6. Applications of Integration:
(u)                                                            (a) Area Between Two Curves:
(v)                                                                    i.
(w)                                                                   ii.
(x)                                                                  iii.
(y) Use trigonometric substitution (R 4:6(b)iii) and           (b) Volume By Slicing:
set x = a tan θ.                                                    i.
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(z) let u = −x                                                         ii.
iii.
4. Partial Fractions:
iv.
(a)                                                                    v.
(b)                                                            (c) Volume By Shells:
(c)                                                                     i.
(d)                                                                    ii.
(e)                                                                  iii.
iv.
(f)
v.
(g)
(d) Arc Length:
(h)
i.
(i)                                                                  ii.
(j)                                                           (e) Surface Area:
(k)                                                                     i.
(l)                                                                   ii.
(m)                                                                   iii.
(n)                                                                   iv.
(o)                                                            (f) Mass From Density:
(p)                                                                     i.
ii.
(q)
iii.
(r)
iv.
(s)                                                            (g) Center of Mass and Centroid:
(t)                                                                   i.
(u)                                                                   ii.
(v)                                                                  iii.

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iv.                                                   (e) Find the Length of the Parametric Arc:
v.                                                         i.
vi.                                                        ii.
vii.
8. Polar Coordinates:
viii.
ix.                                                   (a) Write the Equation in Polar Coordinates:
(h) Fluid Pressure:                                                 i.
i.                                                        ii.
ii.                                                       iii.
(i) Work:                                                     (b) Write the Equation in Cartesian Coordinates:
i.                                                        i.
ii.                                                       ii.
iii.                                                       iii.
iv.                                                   (c) Sketch the Polar Curves:
7. Parametric Curves:                                                    i.
ii.
(a) Express the Parametric Curve by an Equation in
iii.
x and y:
iv.
i.
v.
ii.
vi.
iii.
(d) Calculate the Area enclosed by the Polar Curve:
iv.
v.                                                          i.
ii.
(b) Find a Parametrization x = (t), y = y(t),
t ∈ [0, 1] for                                                iii.
i.                                                       iv.
ii.                                                        v.
iii.                                                   (e) Find the Slope of the Polar Curve:
iv.                                                         i.
(c)                                                                ii.
(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:
i.                                                         i.
ii.                                                        ii.

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