Luca B.
eNotes Educator
Achievements
18
Educator Level
5557
Answers Posted
1244
Answers Bonused
About
I've been a professor of mathematics and computer science since 1992. I am a member of several professional committees, such as the Mathematical Reports Committee and the International Statistical Institute, and I am a reviewer for the journal Mathematical Reviews. I was awarded the Balkan Union Award in 1975 and earned the DAAD scholarship at Bremen University. I am also the author of several scientific and mathematical texts for students.
Earned Badges

Educator of the Month
This badge is awarded to one Educator each month for outstanding contributions to Homework Help. 
eNotes Educator
This badge is awarded to all eNotes Educators. Only official Educators can answer students' questions on our site. Educators are teachers, professional researchers, and scholars who apply to our... 
Hall of Fame
Educators can earn this badge by contributing over 1,000 answers on eNotes. 
Year One Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over one year. 
Year Two Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over two years. 
Year Three Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over three years. 
Year Four Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over four years. 
Year Five Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over five years. 
Year Six Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over six years. 
Year Seven Badge
This badge is awarded once an Educator has been in the eNotes Educator Program for over seven years. 
Quiz Taker
This badge is awarded when an Educator has completed 10 quizzes. 
10K Points Earner
Educators earn points for every question they answer. This Educator has earned over 10,000 points. 
25K Points Earner
Educators earn points for every question they answer. This Educator has earned over 25,000 points. 
50K Points Earner
Educators earn points for every question they answer. This Educator has earned over 50,000 points. 
100K Points Earner
Educators earn points for every question they answer. This Educator has earned over 100,000 points. 
Expert
An expert badge distinguishes Educators who demonstrate strong knowledge in a particular topic, such as Hamlet or Math. It is awarded when an Educator has posted more than 25 answers on a given topic. 
Science Whiz
Bonuses are awarded when an Educator has gone above and beyond and impressed the editorial team by offering an especially lengthy, nuanced, or insightful answer. This badge is given to an Educator... 
Math Whiz
Bonuses are awarded when an Educator has gone above and beyond and impressed the editorial team by offering an especially lengthy, nuanced, or insightful answer. This badge is given to an Educator...
Recent Activity

Answered a Question in Math
You need to use partial fraction decomposition, such that: `(4y^27y12)/(y(y3)(y+2)) = a/y + b/(y3) + c/(y+2)` `4y^27y12= a(y3)(y+2) + by(y+2) + cy(y3)` `4y^27y12=ay^2  ay 6a + by^2 +... 
Answered a Question in Math
You need to use partial fraction decomposition, such that: `(x^2+1)/((x3)(x2)^2) = a/(x3) + b/(x2) + c/((x2)^2)` `x^2+1 = a(x2)^2 + b(x2)(x3) + c(x3)` `x^2+1 = ax^2  4ax + 4a + bx^2  5bx... 
Answered a Question in Math
You may rebuild the structure you find at denominator, such that: `int (x^23x+7)/((x^24x+6)^2)dx = int ((x^24x+6) + x + 1)/((x^24x+6)^2)dx` Separate into two integrals: `int... 
Answered a Question in Math
You need to use reduction formula to integrate the function, such that: `int sec^n x dx = int sec^(n2) x* sec^2 x dx` You need to use integration by parts, such that: `int udv = uv  int vdu` `u =... 
Answered a Question in Math
You need to differentiate the function with respect to x, using product rule such that: `(dy)/(dx) = (d(logx))/(dx)*log x + log x*(d(log x))/(dx)` `(dy)/(dx) = (1/(x*ln 10))*log x + log x*(1/(x*ln... 
Answered a Question in Math
You need to use the following substitution, such that: `x = sin t => dx = cos t dt` `1  x^2 = 1  sin^2 t = cos^2 t` Changing the variable, yields: `int_0^1 x^3*sqrt(1  x^2)dx =... 
Answered a Question in Math
You need to rewrite the expression` t^2  1` , such that: `t^2  1 = t^2(1  (1/t)^2)` You need to use the following substitution, such that: `1/t = sin u => 1/(t^2) dt = cos u du =>... 
Answered a Question in Math
You need to rewrite the expression `t^2  16` , such that: `t^2  16 = t^2(1  (4/t)^2)` You need to use the following substitution, such that: `4/t = sin u => 4/(t^2) dt = cos u du =>... 
Answered a Question in Math
You need to perform the following substitution, such that `36  x^2 = t => 2xdx = dt => xdx = (dt)/2` Replacing the variable yields: `int_0^3 (xdx)/(sqrt(36  x^2)) =... 
Answered a Question in Math
You need to perform the following substitution to solve the integral `sin t = u => cos t dt = du => t = arcsin u` `int_0^(pi/2) (cos t dt)/(sqrt(1 + sin^2 t)) = int_(u_1)^(u_2) (du)/(sqrt(1 +... 
Answered a Question in Math
You need to solve the definite integral, such that: `sqrt(9  25x^2) = sqrt (9(1  25x^2/9))` `sqrt (9(1  25x^2/9)) = 3sqrt(1  25x^2/9)` You need to use the following substitution `(5x/3) = sin u... 
Answered a Question in Math
You need to solve the integral using substitution method, such that: `x^2 = t => 2xdx = dt => xdx = (dt)/2` Replacing x by t, yields: `int x*sqrt(1  x^4) dx = (1/2)int sqrt(1  t^2)*dt` You... 
Answered a Question in Math
You need to use the following trigonometric subtraction formula to prove that `sin(pi/2  x) = cos x` , such that: `sin(a  b) = sin a*cos b  sin b*cos a` Considering` a = pi/2` and ` b = x`... 
Answered a Question in Math
You need to create the volume function, hence, you need to write the formula of volume of the box, such that: V = width x length x height width = length =` 3  2x` height = x `V = (3  2x)(3 ... 
Answered a Question in Math
You need to use the fundamental trigonometric formula `sin^2 x = 1  cos^2 x:` `int sin^2 t*cos^4 t dt = int (1  cos^2 t)*cos^4 t dt` `int (1  cos^2 t)*cos^4 t dt = int cos^4 t dt  int cos^6 t... 
Answered a Question in Math
You need to use integration by parts, such that: `int u dv = uv  int vdu` `u = x => du = dx` `dv = sin^3 x => v = int sin^3 x dx` You need to solve the integral `int sin^3 x dx ` such that:... 
Answered a Question in Math
You need to use the following formula to evaluate the inclination angle theta, such that: `tan theta = m` Replacing`7/9` for m, yields: `tan theta = 7/9 => theta = tan^(1)(7/9) =>... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace `2/x` for x, 3y for y and 5 for n, such that:... 
Answered a Question in Math
You need first to factorize `(4x1)^3` , such that: `(4x1)^3  2(4x1)^4 = (4x1)^3(1  2(4x1))` `(4x1)^3  2(4x1)^4 = (4x1)^3(1  8x + 2) ` `(4x1)^3  2(4x1)^4 = (4x1)^3(38x)` You... 
Answered a Question in Math
You need first to factorize `(x  3)^2` , such that: `2(x3)^4 + 5(x3)^2 = (x3)^2(2(x3)^2 + 5)` You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k))... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 5 for x, `sqrt3*i ` for y and 4 for n, such... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace `2/x` for x, y for y and 4 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 3a for x, 4b for y and 5 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 2x for x, 5y for y and 5 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace `1/x` for x, `y` for y and 5 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace y for x, 2 for y and 5 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace x for x, y for y and 5 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 2x for x, y for y and 3 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace c for x, d for y and 3 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 7a for x, b for y and 3 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace y for x, 4 for y and 3 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace `x^2` for x, `y^2` for y and 4 for n, such... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace x for x, 2y for y and 4 for n, such that:... 
Answered a Question in Math
You need to use the binomial formula, such that: `(x+y)^n = sum_(k=0)^n ((n),(k)) x^(nk) y^k` You need to replace 2 for x, 3i for y and 6 for n, such that: `(23i)^6 = 6C0... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `nCk= (n!)/(k!(nk)!)` You need to replace 12 for n and 0 for k, such that: `12C0= (12!)/(0!(120)!)... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `((n),(k)) = (n!)/(k!(nk)!)` You need to replace 10 for n and 6 for k, such that: `((10),(4)) =... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `((n),(k)) = (n!)/(k!(nk)!)` You need to replace 10 for n and 6 for k, such that: `((10),(6)) =... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `((n),(k)) = (n!)/(k!(nk)!)` You should notice that `((n),(k)) = ((n),(nk))` Hence, `((100),(2)) =... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `((n),(k)) = (n!)/(k!(nk)!)` Replacing 100 for n and 2 for k yields:... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `nCk = (n!)/(k!(nk)!)` Replacing 8 for n and 6 for k yields: `8C6 = (8!)/(6!(86)!) => 8C6 =... 
Answered a Question in Math
You need to evaluate the binomial coefficient, using the next formula, such that: `nCk = (n!)/(k!(nk)!)` Replacing 5 for n and 3 for k yields: `5C3 = (5!)/(3!(53)!) => 5C3 = (5!)/(3!*2!)`... 
Answered a Question in Math
You need to notice that the sequence can be rewritten such that: `3^1/(2^0), (1)^1*(3^2)/(2^1), (1)^2*(3^3)/(2^2),..., (1)^n*(3^(n+1))/(2^n)` Notice that the sequence is a geometric sequence,... 
Answered a Question in Math
You need to use mathematical induction to prove the inequality, hence, you need to perform the following two steps, such that: Step 1: Basis: Prove that the statement holds for n = 1 If x<y... 
Answered a Question in Math
You need to use mathematical induction to prove the inequality, hence, you need to perform the following two steps, such that: Step 1: Basis: Prove that the statement holds for n = 1 `(1+a)^1... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement... 
Answered a Question in Math
You need to use mathematical induction to prove the formula for every positive integer n, hence, you need to perform the two steps of the method, such that: Step 1: Basis: Show that the statement...
Showing 150 of 4964