Question

Find the value of ∫∫xy^7 Cos(x)Cos(y) dxdy.

(a) (7y^6 Cos(y) + 42y^5 Sin(y) + 210y^4 Cos(y) + 840y^3 Sin(y) + 2520y^2 Cos(y) + 5040ySin(y) + 5040Cos(y))(7x^6 Cos(x) + 42x^5 Sin(x) + 210x^4 Cos(x) + 840x^3 Sin(x) + 2520x^2 Cos(x) + 5040xSin(x) + 5040Cos(x))

(b) (y^7 Sin(y) + 7y^6 Cos(y) + 42y^5 Sin(y) + 210y^4 Cos(y) + 840y^3 Sin(y) + 2520y^2 Cos(y) + 5040ySin(y) + 5040Cos(y))(x^7 Sin(x) + 7x^6 Cos(x) + 42x^5 Sin(x) + 210x^4 Cos(x) + 840x^3 Sin(x) + 2520x^2 Cos(x) + 5040xSin(x) + 5040Cos(x))

(c) (y^7 Sin(y) + 42y^5 Sin(y) + 210y^4 Cos(y) + 840y^3 Sin(y) + 2520y^2 Cos(y) + 5040ySin(y) + 5040Cos(y))(x^7 Sin(x) + 7x^6 Cos(x) + 42x^5 Sin(x) + 210x^4 Cos(x) + 840x^3 Sin(x) + 2520x^2 Cos(x) + 5040xSin(x) + 5040Cos(x))

(d) (y^7 Sin(y) + 7y^6 Cos(y) + 42y^5 Sin(y) + 210y^4 Cos(y) + 840y^3 Sin(y) + 2520y^2 Cos(y) + 5040ySin(y) + 5040Cos(y))(x^7 Sin(x) + 7x^6 Cos(x) + 42x^5 Sin(x) + 210x^4 Cos(x) + 840x^3 Sin(x) + 2520x^2 Cos(x) + 5040xSin(x) + 5040Cos(x))

I have been asked this question in a job interview.

I need to ask this question from Double Integrals in section Multiple Integrals of Engineering Mathematics

Answer 1

Right option is (d) (y^7 Sin(y) + 7y^6 Cos(y) + 42y^5 Sin(y) + 210y^4 Cos(y) + 840y^3 Sin(y) + 2520y^2 Cos(y) + 5040ySin(y) + 5040Cos(y))(x^7 Sin(x) + 7x^6 Cos(x) + 42x^5 Sin(x) + 210x^4 Cos(x) + 840x^3 Sin(x) + 2520x^2 Cos(x) + 5040xSin(x) + 5040Cos(x))

The explanation is: Add constant automatically

By, f(x)=\(\int uvdx=\sum_{i=0}^n (-1)^i u_i v^{i+1}\),

Let, u = x^7 and v=Cos(x),

∫x^7 Cos(x) dx=x^7 Sin(x)+7x^6 Cos(x)+42x^5 Sin(x)+210x^4 Cos(x)+840x^3 Sin(x)+2520x^2 Cos(x)+5040xSin(x)+5040Cos(x)

Similarly,

∫y^7 Cos(y) dy=y^7 Sin(y)+7y^6 Cos(y)+42y^5 Sin(y)+210y^4 Cos(y)+840y^3 Sin(y)+2520y^2 Cos(y)+5040ySin(y)+5040Cos(y)

Now,

∫∫xy^7 Cos(x)Cos(y) dxdy=∫y^7 Cos(y) dy∫x^7 Cos(x) dx=(y^7 Sin(y)+7y^6 Cos(y)+42y^5 Sin(y)+210y^4 Cos(y)+840y^3 Sin(y)+2520y^2 Cos(y)+5040ySin(y)+5040Cos(y))(x^7 Sin(x)+7x^6 Cos(x)+42x^5 Sin(x)+210x^4 Cos(x)+840x^3 Sin(x)+2520x^2 Cos(x)+5040xSin(x)+5040Cos(x)).