WMAT1010 Practice Exam
You should attempt the questions in this practice exam before class. It should take approximately two hours
to complete the exam.
1. (a) Evaluate
∫ 6
0
3√
x2+1
dx.
(b) Evaluate
∫
ex sin(ex) dx
2. (a) Evaluate
∫
sin(x) sin(cosx) dx
(b) Evaluate
∫
2x2 + 3x+ 1
(x− 3)(x2 + 2x+ 1)dx using partial fractions.
3. (a) Use four right-hand rectangles to estimate the area under the curve y = 3− x2 for x from 1 to 3.
(b) Use integration by parts to evaluate
∫ pi
0
ex sin(x) dx
4. Use integration by parts to evaluate
∫ pi
0
x2 sin(x) dx
5. Evaluate
∫
1
6 + 6 cos θ + 2 sin θ
dθ using t-substitution. You may find the following values helpful
t = tan
(
θ
2
)
sin(θ) =
2t
1 + t2
cos(θ) =
1− t2
1 + t2
dθ =
2dt
1 + t2
.
6. (a) Solve the following 1st order ODE subject to the boundary condition y(0) = 0
dy
dx
= (y + 1) sin(x)
(b) Solve the following 1st order linear ODE subject to the boundary condition y(1) = 2
y′ +
2y
x
= 3ex
3
7. (a) Solve the following 2nd order ODE subject to the boundary conditions y′(0) = 2 and y(0) = 1
.
y′′ = ex
(b) Solve the following 2nd order ODE.
y′′ + 7y′ + 10y = e−3x
End of Exam
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