Help please - integral evaluation

Question

Please evaluate  

Options:

1. 
2. 
3. x⁴ + 2x + C
4. 

Answer 1

Given:  

Solve

We always simplify or break down each problem so it looks like something we know.

I call this the algebra step.

We know the most used Power Rule formula:

∫ a*xⁿ dx = a * (xⁿ⁺¹) / (n+1) + C

Use algebra to divide x into both numerator parts then simplify.

(x⁴+2x)/x = x⁴/x + 2x/x = x³ + 2

Turn constants, like 2 into 2x⁰ so we can apply the Power Rule.

This helps since 2x⁰ = 2*1 = 2 (if x ≠ 0).

Now integrate each part using the Power Rule formula.

∫ x³ + 2 dx = ∫ x³ + 2x⁰ dx

= ∫ x³ + 2x⁰ dx

= x³⁺¹ / (3+1) + 2x⁰⁺¹ / 0+1 + C

= x⁴ / 4 + 2x¹ / 1 + C

= x⁴ / 4 + 2x + C

SOLUTION:

 = x⁴ / 4 + 2x + C

ANSWER:

2.

Answer 2

Again, the x on the bottom cancels with a factor ox x on the top, so we have the integral of the polynomial, x^3 + 2 dx, which is x^4/4 + 2x + C