Question

\[\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}4x + 3 & , & \text{if }1 \leq x \leq 2 \\3x + 5 & , & \text{if }2 \leq x \leq 4\end{cases}\]

 

Answer 1

 \[\text{We have}\],

\[\int\limits_1^4 f\left( x \right) dx, where\ f\left( x \right) = \begin{cases}4x + 3 & , & \text{if }1 \leq x \leq 2 \\3x + 5 & , & \text{if }2 \leq x \leq 4\end{cases}\]

\[I = \int_1^4 f\left( x \right) d x\]
\[ \Rightarrow I = \int_1^2 f\left( x \right) d x + \int_2^4 f\left( x \right) d x ..............\left[ \text{Additive property} \right]\]
\[ \Rightarrow I = \int_1^2 \left( 4x + 3 \right) dx + \int_2^4 \left( 3x + 5 \right) dx\]
\[ \Rightarrow I = \left[ 2 x^2 + 3x \right]_1^2 + \left[ \frac{3 x^2}{2} + 5x \right]_2^4 \]
\[ \Rightarrow I = 8 + 6 - 2 - 3 + 24 + 20 - 6 - 10\]
\[ \Rightarrow I = 37\]