Evaluate the following definite integrals

Question

(a) ׬0
1
𝑥
4 + 2𝑥3 + 4𝑥 + 10 𝑑𝑥

Answer 1

To evaluate this definite integral, we can first simplify the integrand by factoring it:

x^4 + 2x^3 + 4x + 10 = x^4 + 2x^3 + 4x + 10
= x^3(x + 2) + 4(x + 2)
= (x^3 + 4)(x + 2)

Now, the integral becomes:

∫(x^3 + 4)(x + 2) dx

Expanding the integrand:

= ∫x^4 + 2x^3 + 4x + 8 dx

Integrating term by term:

= x^5/5 + x^4/2 + 2x^2 + 8x + C

Now we can evaluate the definite integral:

∫[0,1] (x^4 + 2x^3 + 4x + 10) dx = [1^5/5 + 1^4/2 + 2(1)^2 + 8(1)] - [0^5/5 + 0^4/2 + 2(0)^2 + 8(0)]
= 1/5 + 1/2 + 2 + 8
= 11.5

Thus, the value of the definite integral is 11.5.