Evaluate the following integrals.(b)∫ 21xe3x dx (c)∫ cos(√x)√x dx (d)∫ x2 + 3x + 3x + 1 dx

Question

(b) ∫ 21xe^{3x} dx
(c) ∫ cos(√x)√x dx
(d) ∫ (x^2 + 3x + 3x + 1) dx

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Sure, let's solve these integrals one by one.

(b) ∫ 21xe^3x dx

This is an example of integration by parts, which is given by ∫udv = uv - ∫vdu. We can let u = x and dv = 21e^3x dx. Then, du = dx and v = (7/3)e^3x.

So, ∫udv = uv - ∫vdu = x*(7/3)e^3x - ∫(7/3)e^3x dx = (7/3)xe^3x - (7/9)e^3x + C.

This integral is a bit

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