Calculus

### **Step-by-Step Solution & Mathematical Derivation (GATE BT 2018 Q.51)** **1. Governing Concept & Definition**: - **Topic**: Engineering Mathematics & Definite Integrals - Evaluating the polynomial integral: $\int_0^1 (x + 3)(x + 1)\,dx$ **2. Detailed Step-by-Step Calculation**: - **Expand the integrand**: $(x + 3)(x + 1) = x^2 + 4x + 3$ - **Integrate term by term**: $\int_0^1 (x^2 + 4x + 3)\,dx = \left[ \frac{x^3}{3} + 2x^2 + 3x \right]_0^1$ - **Evaluate at limits**: $\left( \frac{1^3}{3} + 2(1)^2 + 3(1) \right) - 0 = \frac{1}{3} + 2 + 3 = \frac{1}{3} + 5 = \frac{16}{3} \approx 5.333 \approx 5.33$ **3. Final Answer**: - **Calculated Value**: **5.33** - **Official GATE Accepted Range**: **5.30 to 5.34**