Question

$\int_{a}^{b} x^{3} d x =_________$

$\frac{b^{3} - a^{3}}{3}$
$\frac{b^{4} - a^{4}}{4}$
$\frac{b^{2} - a^{2}}{2}$
$0$

Answer: $\frac{b^{4} - a^{4}}{4}$

Solution :-

\begin{aligned} We know that \int_{a}^{b} f (x) d x & = & F (x) ∣_{a}^{b} \\ = & F (b) - F (a) \cdot \end{aligned}

Now, according to question

\begin{aligned} \int_{a}^{b} x^{3} d x & = & \frac{x^{3 + 1}}{3 + 1} |_{a}^{b} \\ = & \frac{x^{4}}{4} |_{a}^{b} \\ = & \frac{b^{4}}{4} - \frac{a^{4}}{4} \\ = & \frac{b^{4} - a^{4}}{4} \cdot \end{aligned}

So, correct answer is option-2, $\to \frac{b^{4} - a^{4}}{4}$

#Differentiation #BSEB PYQ 2017 #Mathematics #BSEB PYQs