Question

Find the value of ∫x^7 Cos(x) dx.

(a) 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)

(b) 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) 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) 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) + 10080Cos(x)

This question was posed to me in homework.

The origin of the question is Improper Integrals in division Integral Calculus of Engineering Mathematics

Answer 1

The correct choice is (a) 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: 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)