For each term the aim is to raise the power of x by 1 and divide by the new power.

For this question, each part of the expression can be looked at seperately to make things a bit easier:

∫(1 + 3√x + 5x)dx = ∫1dx + ∫3√xdx + ∫5xdx

The first part of the expression can be looked at as 1x⁰, so the integral of this is 1x = x

The second part is a bit more difficult as the power of x isnt a whole number so it can be written as 3x^1/2, the integral of this being 3x^3/2*(2/3) = 2x^3/2, (the 2/3 comes from dividing by the new power).

Finally the integral of 5x is easier as the power of x is a whole number and so is easily calculated as 5/2*x².

Then finally recombining the three part the final answer is:

∫(1 + 3√x + 5x)dx = x + 2x^3/2 + (5/2)x²+ c

(c is constant and can take any value, this isnt a majorly important part of the question)

x^3 + 3x^2 + 2x + 12

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∫(1 + 3√x + 5x)dx

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