Q.Evaluate the definite integral: $\int_1^2 (4x^3 - 5x^2 + 6x + 9) \ dx$

Concept: Definite Integral — Power Rule & Linear Combination

We integrate term-by-term using $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ and then evaluate from $1$ to $2$.

Step 1: Find the antiderivative

Step 2: Evaluate at the limits

At $x = 2$:

At $x = 1$:

Step 3: Subtract

Step-by-step solution

1. Find the antiderivative.

2. Evaluate at the upper limit $x=2$.

3. Evaluate at the lower limit $x=1$.

4. Subtract.

Method: Definite integral of a polynomial (term-by-term power rule)

Integrate each power of $x$ separately, then evaluate $F(b)-F(a)$.

Steps

Step 1: Apply $\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}$ to every term.

For $4x^{3}-5x^{2}+6x+9$: $F(x)=x^{4}-\dfrac{5x^{3}}{3}+3x^{2}+9x$.

Step 2: Form the evaluation bracket $\big[F(x)\big]_{a}^{b}$.

Step 3: Substitute the upper and lower limits and subtract, keeping fractions exact.

Step 4: Combine to a single value. No $+C$ for a definite integral; a common denominator tidies the fractions.

Common Mistakes

Mistake 1: Antiderivative of $-5x^{2}$ taken as $-5x^{3}$ (forgetting $\div 3$).

Why it's wrong: $\int -5x^{2}\,dx=-\dfrac{5x^{3}}{3}$. Correct approach: divide by the new exponent.

Mistake 2: Antiderivative of the constant $9$ dropped.

Why it's wrong: $\int 9\,dx=9x$, a real contribution. Correct approach: integrate constants to $9x$.

Mistake 3: Arithmetic slip in $F(2)-F(1)$ with mixed fractions.

Why it's wrong: careless subtraction gives a wrong number. Correct approach: convert to a common denominator, e.g. $\dfrac{98}{3}-\dfrac{34}{3}=\dfrac{64}{3}$.