If `a=(3t^2+2t+1)` `m/s^2` is the expression according to which the acceleration of a particle varies. Then-
The expression for instantaneous velocity at any time `t` will be (if the particle was initially at rest)-

Options

A. `t^3+2t+1`
B. `t^3+t+1`
C. `t^3+t^2+1`
D. `t^3+t^2+t+C`

Explanation

To solve the problem, we need to find the expression for instantaneous velocity \( v(t) \) of a particle given its acceleration \( a(t) = 3t^2 + 2t + 1 \, \text{m/s}^2 \) and that the particle was initially at rest.

### Step-by-Step Solution:

1. **Understand the relationship between acceleration and velocity**:
Acceleration \( a \) is defined as the rate of change of velocity \( v \) with respect to time \( t \). Mathematically, this is expressed as:
\[ a = \frac{dv}{dt} \]

2. **Set up the equation**:
From the definition of acceleration, we can rearrange the equation:
\[ dv = a \, dt \]
Substituting the given expression for acceleration:
\[ dv = (3t^2 + 2t + 1) \, dt \]

3. **Integrate both sides**:
To find the velocity, we integrate both sides. The limits for \( v \) will be from \( 0 \) (initial velocity) to \( v \) (velocity at time \( t \)), and for \( t \) from \( 0 \) to \( t \):
\[ \int_0^v dv = \int_0^t (3t^2 + 2t + 1) \, dt \]

4. **Calculate the left side**:
The left side integrates to:
\[ v - 0 = v \]

5. **Calculate the right side**:
Now we need to compute the integral on the right side:
\[ \int (3t^2 + 2t + 1) \, dt = \int 3t^2 \, dt + \int 2t \, dt + \int 1 \, dt \]
- The integral of \( 3t^2 \) is \( t^3 \).
- The integral of \( 2t \) is \( t^2 \).
- The integral of \( 1 \) is \( t \).

Therefore, we have:
\[ \int (3t^2 + 2t + 1) \, dt = t^3 + t^2 + t \]

6. **Evaluate the definite integral**:
Now we evaluate from \( 0 \) to \( t \):
\[ v = \left[ t^3 + t^2 + t \right]_0^t = (t^3 + t^2 + t) - (0 + 0 + 0) = t^3 + t^2 + t \]

7. **Final expression for velocity**:
Thus, the expression for instantaneous velocity at any time \( t \) is:
\[ v(t) = t^3 + t^2 + t \, \text{m/s} \]

### Final Answer:
The expression for instantaneous velocity at any time \( t \) is:
\[ v(t) = t^3 + t^2 + t \, \text{m/s} \]