Applications of integration

Differentiate y=5x(e3x) with respect to x and hence find ∫x(e3x) dx

Solution: 1/3(x)(e3x) – 1/9(e3x) + C

Y = 5x e3x

dy/dx = 5x(e3x)(3) + 5 e3x

dy/dx = 15x(e3x) + 5(e3x)

∫15x(e3x) + 5(e3x) dx = 5x(e3x)

∫15x(e3x) dx + (5(e3x))/3 = 5x(e3x)

15∫x(e3x) dx = 5x(e3x) – 5/3(e3x)

∫x(e3x) dx = 1/3(x)(e3x) – 1/9(e3x) + C

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