Exercise 1:
Calculate
by looking for an intermediate integral of the form
that will easily fit in.
We consider the integral:
Let’s calculate:
so
Exercise 2:
Calculate these integrals by integrating by partials:
A. .
Let u=x u’=1
and
Exercise 3:
Let be the function defined on
by
.
What is the derivative of on
?
Exercise 4:
Exercise 5:
Exercise 6:
Consider the sequence defined, for any natural number n , by :
1.
a. Show that .
b. Calculate , deduce
.
2. Show that, for any natural number n, .
3.
a. Show that, for any non-zero natural number n,
b. Deduce that, for any non-zero natural number n,
4. Determine the limit of the sequence .
Exercise 7:
Exercise 8:
Exercise 9:
Calculate:
Let’s ask:
and
Exercise 10:
a.
b.
c.
d.
d.
e.
f.
g.
h.
i.
j.
k.
l.
m.
Exercise 12:
We consider three real numbers a,b,c such that, for all :
.
.
.
By identification, we obtain the system :
so
2. Let .
a. Calculate
.
b. Let f be the function defined on by
By integrating by parts, calculate f(X) as a function of X .
Let’s ask:
u(x)= lnx then
then
So:
c. gold :
therefore:
Exercise 14:
Calculate the proposed integral:
a.
b.
c.
d.
e.
f.
g.
h.
Exercise 15:
Either
Exercise 16:
We consider
and
a.
b.
Exercise 17:
because the function is odd on the interval [-5;5] centered in 0.
Answers to the exercises on integrals in high school.
After having consulted the answers to these exercises on integral calculus in the final year of high school, you can return to the exercises in the final year of high school.
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