If the area bounded by the curve `y=x^2+1` and the tangents to it drawn from the origin is `A ,` then the value of `3A` is__-

Explanation

To solve the problem, we need to find the area \( A \) bounded by the curve \( y = x^2 + 1 \) and the tangents drawn from the origin, and then compute \( 3A \).

### Step-by-Step Solution:

1. **Identify the curve and the point of tangency**:
The curve is given by \( y = x^2 + 1 \). We need to find the points on this curve where the tangents drawn from the origin touch the curve.

2. **Find the slope of the tangent**:
Let the point of tangency on the curve be \( (x, x^2 + 1) \). The slope of the tangent line at this point is given by the derivative of the curve:
\[ \frac{dy}{dx} = 2x \]
The slope of the line joining the origin \( (0, 0) \) to the point \( (x, x^2 + 1) \) is:
\[ \text{slope} = \frac{x^2 + 1 - 0}{x - 0} = \frac{x^2 + 1}{x} \]

3. **Set the slopes equal**:
For the line to be tangent, the slopes must be equal:
\[ 2x = \frac{x^2 + 1}{x} \]
Multiplying through by \( x \) (assuming \( x \neq 0 \)):
\[ 2x^2 = x^2 + 1 \]
Rearranging gives:
\[ x^2 - 1 = 0 \]
Thus, \( x^2 = 1 \) which implies \( x = 1 \) or \( x = -1 \).

4. **Find the equations of the tangents**:
For \( x = 1 \):
\[ y - (1^2 + 1) = 2(1)(x - 1) \implies y - 2 = 2(x - 1) \implies y = 2x \]
For \( x = -1 \):
\[ y - ((-1)^2 + 1) = -2(-1)(x + 1) \implies y - 2 = -2(-1)(x + 1) \implies y = -2x + 4 \]

5. **Set up the integral for the area**:
The area \( A \) between the curves \( y = x^2 + 1 \) and the tangents \( y = 2x \) and \( y = -2x + 4 \) from \( x = 0 \) to \( x = 1 \) is given by:
\[ A = \int_0^1 [(x^2 + 1) - 2x] \, dx + \int_1^2 [(-2x + 4) - (x^2 + 1)] \, dx \]

However, since the area is symmetric, we can calculate just one part and double it:
\[ A = 2 \int_0^1 [(x^2 + 1) - 2x] \, dx \]

6. **Calculate the integral**:
\[ A = 2 \int_0^1 (x^2 - 2x + 1) \, dx = 2 \int_0^1 (x - 1)^2 \, dx \]
Now, calculating the integral:
\[ = 2 \left[ \frac{(x - 1)^3}{3} \right]_0^1 = 2 \left[ \frac{(1 - 1)^3}{3} - \frac{(0 - 1)^3}{3} \right] = 2 \left[ 0 - \left(-\frac{1}{3}\right) \right] = 2 \cdot \frac{1}{3} = \frac{2}{3} \]

7. **Calculate \( 3A \)**:
\[ 3A = 3 \cdot \frac{2}{3} = 2 \]

### Final Answer:
The value of \( 3A \) is \( 2 \).