Integral ∫$(5x^3 + 7x^2 − 2x + 5)$dx

To solve the integral $\int (5x^3 + 7x^2 - 2x + 5) \, dx$, we apply the Power Rule for Integration.

1. The Concept

The Power Rule states that for any term $ax^n$:

(where $n \neq -1$ and $C$ is the constant of integration).

2. Step-by-Step Solution

We integrate each term in the expression individually:

• For $5x^3$: Add 1 to the exponent (3 + 1 = 4) and divide by the new exponent: $\int 5x^3 \, dx = \frac{5x^4}{4}$

• For $7x^2$: Add 1 to the exponent (2 + 1 = 3) and divide by the new exponent: $\int 7x^2 \, dx = \frac{7x^3}{3}$

• For $-2x$ (which is $-2x^1$): Add 1 to the exponent (1 + 1 = 2) and divide by the new exponent: $\int -2x^1 \, dx = \frac{-2x^2}{2} = -x^2$

• For $5$ (which is $5x^0$): Add 1 to the exponent (0 + 1 = 1) and divide by the new exponent: $\int 5 \, dx = 5x$

• Add the constant of integration ($C$): Since this is an indefinite integral, we must add $C$ at the end.

3. Combining the results

Putting it all together, we get:

Comparing this result to your options, it matches Option B.

Exam Tip

When integrating polynomials, always remember to check your work by differentiating your answer. If you differentiate $\frac{5x^4}{4} + \frac{7x^3}{3} - x^2 + 5x + C$, you will arrive back at the original expression $(5x^3 + 7x^2 - 2x + 5)$. This is a great way to verify your answer during the JAMB exam!