Question

If \[ \int_0^1 (3x^2+2x+1)\,dx=k, \] then the value of \(k\) is

• A. \(2\)
• B. \(3\)
• C. \(4\)
• D. \(5\)

Hint:

For polynomial integrals, integrate each term separately and then substitute the upper and lower limits carefully. This avoids common arithmetic mistakes.

Answer 1

The Correct Option is B

Step 1: Understand the task.
We must find the number $k=\displaystyle\int_0^1 (3x^2+2x+1)\,dx$. A definite integral measures the area-type total of the function between the limits.

Step 2: Recall the basic rule.
The power rule for integration is $\displaystyle\int x^n\,dx=\dfrac{x^{n+1}}{n+1}$. We use it on each term.

Step 3: Integrate term by term.
\[ \int 3x^2\,dx=x^3,\quad \int 2x\,dx=x^2,\quad \int 1\,dx=x. \] So the antiderivative is $F(x)=x^3+x^2+x$.

Step 4: Apply the limits.
By the Fundamental Theorem, $k=F(1)-F(0)$.

Step 5: Compute each value.
At $x=1$: $1^3+1^2+1=3$. At $x=0$: $0+0+0=0$.

Step 6: Subtract.
\[ k=3-0=3. \] This is option 2.
\[ \boxed{3} \]