Midterm Practice Problems
Disclaimer: I in no way suggest that this is comprehensive. Students should review their notes,
assignments and sample problems in conjunction with these practice problems. Sample midterms
are on the wiki.
For reference, you may need some or all of the following formulas
n
X
i=1
n(n + 1)
i=
2
n
X
i=1
n(n + 1)(2n + 1)
i =
6
2
n
X
i=1
i3 =
n2 (n + 1)2
4
1) Using the definition of an integral (with right endpoints in the Riemann sums), evaluate
R4
i. 0 3dx
R1
ii. 0 (2x + 3)dx
R
2) Integrate sec3 (x)dx (Hint: Try by parts splitting up the secant terms)
3) Integrate all of the below.
R2
2
(i) 1 1+u
3 du
R 2u
(ii) (u − 1)2 du
R
(iii) arctan(x)dx
R
(iv) tan(x)dx
Rπ
(v) 04 sec2 (t)dt
R
(vi) eu sin(u)du
R
(vii) ln(x)
dx
x2
R 2 x−1
(viii) 1 e x2 dx
R 1
(ix) 2x
dx
R
1
dx
(x) √1−x
2
Rπ
(xi) 0 sec(x) tan(x)dx
R
(xii) 10u du
R1
(xiii) −1 x2 sin(x)dx
R1
(xiv) 0 t24+1 dt
R
(xv) sec(t)dt
R4
(xvi) 1 x13 dx
R t9
(xvii) 1+t
20 dt
R
(xviii) x ln(x)dx
R1
(xix) −1 |x3 − x|dx
R x2
(xx) 1+x
2 dx
Rπ
(xxi) 0 sin(x)dx
R 3
(xxii) t ln(t)
dt
Rπ
(xxiii) 0 cos(x)dx
R
(xxiv) x2 ex dx
R 3 +1
(xxv) 4x
dx
x4 +x
(xxvi)
R
(xxvii)
R
(xxviii)
(xxix)
(xxx)
(xxxi)
(xxxii)
(xxxiii)
(xxxiv)
(xxxv)
(xxxvi)
(xxxvii)
(xxxviii)
(xxxix)
(xl)
(xli)
(xlii)
(xliii)
(xliv)
(xlv)
(xlvi)
(xlvii)
(xlviii)
(xlix)
(l)
(li)
(lii)
(liii)
(liv)
(lv)
(lvi)
(lvii)
(lviii)
(lix)
(lx)
(lxi)
(lxii)
(lxiii)
(lxiv)
(lxv)
√
4x2 x3 + 2dx
ln(t)dt
R3 2
|x − 4|dx
R−5
2t log3 (t)dt
R 10
(2x3 + 5x7 + 11x13 + 17x19 + 23x29 )dx
R−10
6dt
R ln(3) et
0
et +1 dt
R 3
x cos(4x4 + 10)dx
R3 2
(x + x + 1)dx
R−1 x
dx
1+x4
R
2
sec (t)dt
R
x sin(x)dx
R e3
2
2 (ln(x)) dx
Re
u arctan(u)du
R 1+x
dx
1+x2
R −x
xe dx
R
cos(x)esin(x) dx
R
cos(x) sin(x)dx
R 2
x cos(x) dx
R x
e sin(x) dx
R
tan5 (x) dx
R x+27
dx
x2 −9
R x+27
dx
x2 +9
p
R
sec2 (x) 5 + tan(x) dx
R
cos2 (x) dx
R 2√
x 4 − x2 dx
R 2√
x 4 + x2 dx
R 2√
x x2 − 4 dx
R 2x+1
x(1−x) dx
R
(csc(x) + cot(x))2 dx **
R
sin10 (x) cos5 (x) dx
R
cos2 (x) dx
R
sin2 (x) dx
R 2
ln (x) dx
R x
5 dx
R 7x−26
dx
x2 −6x−16
R tan(x) 3
dx
cos(x)
R1 1
−1 x dx
R1
−1 x dx
R1
1
−1 x0.999 dx
2
√ √
e x x dx
R√
(lxvii)
x2 + 6x dx
R
(lxviii) tan4 (x) sec4 (x) dx
R1
1
(lxix) 1/2 √2x−x
dx
2
R
(lxx) ex √e12x −9 dx
R
2 (x)
(lxxi) tan2sec
dx
(x)−tan(x)
R 1
(lxxii) x4 −16 dx
R
(lxxiii) x41+4 dx
R
(lxxiv) x21+4 dx
R
(lxxv) √x(√1x+1) dx
R
(lxxvi) sec3 (x) dx **
R
1
(lxxvii) x√1+x
dx
2
R2 x
(lxxviii) −1 |e − 1|
R8 4
(lxxix) 6 (x−6)
3 dx
R1
(lxxx) 0 arcsin(x) dx
R
(lxxxi) arccos(x) dx
R1
(lxxxii) 0 x35 dx
R 4x2 −2x
(lxxxiii) (x−1)(x
2 +1) dx
R
(lxxxiv) tan(x) sec3 (x) dx
R
1
dx
(lxxxv) √x2 +2x+2
R
(lxxxvi) cos(πx)) dx
R
(lxxxvii) sin(x) cos5 (x) dx
R 1
(lxxxviii) x√
dx
x
R
(lxxxix) sin5 cos2 (x) dx
R 4 +3x2 +1
(xc) xx5 +5x
3 +5x dx
(lxvi)
R
4) Find the centroid of the region bounded by
(i) x + y = 2, x = y 2 .
(ii) y = 1 + sin(x) between x = −π/2 and x = π/2 and the x-axis.
(iii) y = 5x between y = −1 and x = 2.
3