Question

\[\int\limits_1^4 f\left( x \right) dx, where f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]

Answer 1

We have,

\[\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}7x + 3 & , & \text{if }1 \leq x \leq 3 \\ 8x & , & \text{if }3 \leq x \leq 4\end{cases}\]

LaTeX

\[I = \int_1^4 f\left( x \right) d x\]
\[ \Rightarrow I = \int_1^3 f\left( x \right) d x + \int_3^4 f\left( x \right) d x ...................\left[ \text{Additive property} \right]\]
\[ \Rightarrow I = \int_1^3 \left( 7x + 3 \right) d x + \int_3^4 8x d x\]
\[ \Rightarrow I = \left[ \frac{7 x^2}{2} + 3x \right]_1^3 + \left[ 4 x^2 \right]_3^4 \]
\[ \Rightarrow I = \frac{63}{2} + 9 - \frac{7}{2} - 3 + 64 - 36\]
\[ \Rightarrow I = \frac{56}{2} + 34\]
\[ \Rightarrow I = 62\]