Integration 1

Differentiate y = 5xe2x+1 with respect to x and hence, find ∫x(e2x+1) dx.

Solution: 1/2(x)(e2x+1) – 1/4(e2x+1) + C

dy/dx = 5x(e2x+1)(2) + 5(e2x+1)

          =10x(e2x+1) + 5(e2x+1)

5x(e2x+1) = ∫10x(e2x+1) + 5(e2x+1) dx

5x(e2x+1) = 10∫x(e2x+1) dx + 5∫(e2x+1) dx

10∫x(e2x+1) dx + 5/2(e2x+1) = 5x(e2x+1)

10∫x(e2x+1) dx = 5x(e2x+1) -5/2(e2x+1)

∫x(e2x+1) dx = 1/2(x)(e2x+1) – 1/4(e2x+1) + C

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