Compute the value of the integral :I = Z e 1 x (x2+ 1)2lnxdx

L1 - Bilingual Group - MS2 - 2008/2009 - 20 May 2009 - Session 1

Examination in Mathematics for Sciences (MS2)

Duration: 3 hours Documents and any type of calculators are not allowed

Exercice 1 : 1. Decompose into simple elements onC, after that onR, the rational fraction 1 X(X²+ 1). 2. Compute the value of the integral :I =

Z e

1

x

(x²+ 1)²lnxdx. (Hint : rst, integration by parts) Exercice 2 : 1. CalculateI =

Z π

0

sinx 1 + cos²xdx. 2. Letf(x) = xsinx

1 + cos²x. Show the identity :f(x) +f(π−x) = πsinx 1 + cos²x. 3. Deduce from the previous questions the value ofJ =

Z π

0

xsinx 1 + cos²xdx.

4. By computingJdirectly by integration by parts, infer the value of the integralK = Z π

0

Arctan(cosx)dx.

Exercice 3 : Leta, b∈R,a < b. LetIm,n= Z b

a

(x−a)^m(b−x)ⁿdx, with(m, n)∈N². 1. Show thatI_m,n = m

n+ 1Im−1,n+1,∀(m, n)∈N^∗×Nand I_m,n =− n

m+ 1Im+1,n−1,∀(m, n)∈N×N^∗. 2. ComputeI_0,m+n and deduce from the previous questions that I_m,n = m!n!(b−a)^m+n+1

(m+n+ 1)! . Exercice 4 : Let the functionf :R² →Rbe dened by f(x, y) =x³(1−4y²) +y².

1. Compute its rst order partial derivatives (denoted as∂_xf ≡ ^∂f_∂x and ∂_yf ≡ ^∂f_∂y).

2. Show thatf has only 3 critical points, namely (0,0),( 1

√3

4,1

2)and ( 1

√3

4,−1 2).

3. Compute in each point (x, y) ∈ R the second order partial derivatives of f (denoted by ∂xxf ≡ ^∂_∂x^2f₂,

∂xyf ≡ _∂x∂y^∂2f ,∂yxf ≡ _∂y∂x^∂2f ,∂yyf ≡ ^∂_∂y^2f₂).

4. Compute for each(x, y)∈R, the expression of the determinantdetH(x, y) =

∂_xxf(x, y) ∂_xyf(x, y)

∂_yxf(x, y) ∂_yyf(x, y) . 5. Give the values ofdetH(x_i, y_i) where(x_i, y_i),i= 1,2,3, are the critical points of f.

6. According to the values found at the preceding question, discuss and decide (if possible) on the nature of the critical points (extremum, saddle points ?).

7. Calculatef(0,0). Write the expression off(x, y) in polar coordinates and use it in order to study the sign of f in an arbitrarily small ball centered on (0,0). Conclude : isf(0,0)or not a local extremum off?

Exercice 5 : Letf :I →R(with I interval of R) be a dierentiable function and let F be a primitive of f. Let (E) be the dierential equation (with unknown functiony:I →R) :y⁰⁰+f y⁰+f⁰y= 0.

1. Show that (E) is equivalent to the sentence : there is someC∈Rsuch that (e_C) : y⁰+f y=C onI. 2. Denote by (e⁰_C) the homogenous (null right hand side) equation associated to (eC). Give the expression

of the general solutiony₀ of (e⁰_C) (as an expression depending on F).

3. Use the variation of the constant method in order to nd an expression for a particular solution yp of (eC) (as an expression depending on F). Deduce from the previous questions the general solution of (eC).

4. Give the general solution of (E) (as an expression depending onF).

5. Forx∈I =

−^π₂,^π₂

let f(x) = tanx

3 + cos(2x). Compute a primitiveF of f.

6. Deduce, for this case, the general solution of (E) in terms of a primitiveGofg(x) = √⁴

2 + tan²x(one shall not try to computeG explicitely).

Note : In the exercises 4 and 5 the last question is lightly harder than the others.