The value of `int_(0)^(2) 3x^(2)dx+int_(0)^(pi//2) sin x dx` is :-

Options

A. 9
B. 7
C. 8
D. 13

Explanation

To solve the problem, we need to evaluate the following expression:

\[ \int_{0}^{2} 3x^{2} \, dx + \int_{0}^{\frac{\pi}{2}} \sin x \, dx \]

### Step 1: Evaluate the first integral \(\int_{0}^{2} 3x^{2} \, dx\)

1. **Find the antiderivative of \(3x^{2}\)**:
\[ \int 3x^{2} \, dx = x^{3} + C \]

2. **Apply the limits from 0 to 2**:
\[ \left[ x^{3} \right]_{0}^{2} = (2^{3}) - (0^{3}) = 8 - 0 = 8 \]

### Step 2: Evaluate the second integral \(\int_{0}^{\frac{\pi}{2}} \sin x \, dx\)

1. **Find the antiderivative of \(\sin x\)**:
\[ \int \sin x \, dx = -\cos x + C \]

2. **Apply the limits from 0 to \(\frac{\pi}{2}\)**:
\[ \left[ -\cos x \right]_{0}^{\frac{\pi}{2}} = \left(-\cos\left(\frac{\pi}{2}\right)\right) - \left(-\cos(0)\right) = (0) - (-1) = 1 \]

### Step 3: Combine the results of both integrals

Now, we can add the results from both integrals:
\[ 8 + 1 = 9 \]

### Final Answer
The value of the expression \(\int_{0}^{2} 3x^{2} \, dx + \int_{0}^{\frac{\pi}{2}} \sin x \, dx\) is:
\[ \boxed{9} \]