Question

Find the value of ∫x^3 Sin(x)dx.

(a) x^3 Cos(x) + 3x^2 Sin(x) + 6xCos(x) – 6Sin(x)

(b) – x^3 Cos(x) + 3x^2 Sin(x) – 6Sin(x)

(c) – x^3 Cos(x) – 3x^2 Sin(x) + 6xCos(x) – 6Sin(x)

(d) – x^3 Cos(x) + 3x^2 Sin(x) + 6xCos(x) – 6Sin(x)

The question was asked in exam.

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

Answer 1

Right option is (d) – x^3 Cos(x) + 3x^2 Sin(x) + 6xCos(x) – 6Sin(x)

Best explanation: Add constant automatically

Let f(x) = x^3 Sin(x)

∫x^3 Sin(x)dx = – x^3 Cos(x) + 3∫x^2 Cos(x)dx

∫x^2 Cos(x)dx = x^2 Sin(x) – 2∫xSin(x)dx

∫xSin(x)dx = – xCos(x) + ∫Cos(x)dx = – xCos(x) + Sin(x)

=> ∫x^3 Sin(x)dx = – x^3 Cos(x) + 3[x^2 Sin(x) – 2[ – xCos(x) + Sin(x)]]

=> ∫x^3 Sin(x)dx = – x^3 Cos(x) + 3x^2 Sin(x) + 6xCos(x) – 6Sin(x).