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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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