What is `int_(-0)^pi (x^3+sinx)` equal to

Options

A. a
B. 2a
C. 0
D. 1

Answer

To solve the integral \( \int_{0}^{\pi} (x^3 + \sin x) \, dx \), we can break it down into two parts:

1. **Calculate the integral of \( x^3 \) from 0 to \( \pi \)**
2. **Calculate the integral of \( \sin x \) from 0 to \( \pi \)**
3. **Add the results of both integrals**

### Step 1: Calculate \( \int_{0}^{\pi} x^3 \, dx \)

The formula for the integral of \( x^n \) is:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
For \( n = 3 \):
\[ \int x^3 \, dx = \frac{x^{4}}{4} + C \]

Now, we evaluate the definite integral:
\[ \int_{0}^{\pi} x^3 \, dx = \left[ \frac{x^4}{4} \right]_{0}^{\pi} = \frac{\pi^4}{4} - \frac{0^4}{4} = \frac{\pi^4}{4} \]

### Step 2: Calculate \( \int_{0}^{\pi} \sin x \, dx \)

The integral of \( \sin x \) is:
\[ \int \sin x \, dx = -\cos x + C \]

Now, we evaluate the definite integral:
\[ \int_{0}^{\pi} \sin x \, dx = \left[ -\cos x \right]_{0}^{\pi} = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 = 2 \]

### Step 3: Combine the results

Now, we add the results of both integrals:
\[ \int_{0}^{\pi} (x^3 + \sin x) \, dx = \int_{0}^{\pi} x^3 \, dx + \int_{0}^{\pi} \sin x \, dx = \frac{\pi^4}{4} + 2 \]

Thus, the final result is:
\[ \int_{0}^{\pi} (x^3 + \sin x) \, dx = \frac{\pi^4}{4} + 2 \]