Short Question 5 – 7 – user's Blog!

Question 5:

$\begin{array}{l} Given \int (6 x^{2} + 1) d x = m x^{3} + x + c, \\ where m and c are constants, find \\ (a) the value of m . \\ (b) the value of c if \int (6 x^{2} + 1) d x = 13 when x = 1. \end{array}$

Solution:
(a)

$\begin{array}{l} \int (6 x^{2} + 1) d x = m x^{3} + x + c \\ \frac{6 x^{3}}{3} + x + c = m x^{3} + x + c \\ 2 x^{3} + x + c = m x^{3} + x + c \\ Compare the both sides, \\ ∴ m = 2 \end{array}$

(b)

$\begin{array}{l} \int (6 x^{2} + 1) d x = 13 when x = 1. \\ 2 {(1)}^{3} + 1 + c = 13 \\ 3 + c = 13 \\ c = 10 \end{array}$

Question 6:

$\begin{array}{l} It is given that \int_{5}^{k} g (x) d x = 6, and \int_{5}^{k} [g (x) + 2] d x = 14, \\ find the value of k . \end{array}$

Solution:

$\begin{array}{l} \int_{5}^{k} [g (x) + 2] d x = 14 \\ \int_{5}^{k} g (x) d x + \int_{5}^{k} 2 d x = 14 \\ 6 + {[2 x]}_{5}^{k} = 14 \\ 2 (k - 5) = 8 \\ k - 5 = 4 \\ k = 9 \end{array}$

Question 7:

$Given \int_{k}^{2} (4 x + 7) d x = 28, calculate the possible value of k .$

Solution:

$\begin{array}{l} \int_{k}^{2} (4 x + 7) d x = 28 \\ {[2 x^{2} + 7 x]}_{k}^{2} = 28 \\ 8 + 14 - (2 k^{2} + 7 k) = 28 \\ 22 - 2 k^{2} - 7 k = 28 \\ 2 k^{2} + 7 k + 6 = 0 \\ (2 k + 3) (k + 2) = 0 \\ k = - \frac{3}{2} or k = - 2 \end{array}$