JAMB - Mathematics (2014 - No. 33)
Evaluate \(\int (2x + 3)^{\frac{1}{2}} \delta x\)
\(\frac{1}{12} (2x + 3)^6 + k\)
\(\frac{1}{3} (2x + 3)^{\frac{1}{2}} + k\)
\(\frac{1}{3} (2x + 3)^{\frac{3}{2}} + k\)
\(\frac{1}{12} (2x + 3)^{\frac{3}{4}} + k\)
Explanation
\(\int (2x + 3)^{\frac{1}{2}} \delta x\)
let u = 2x + 3, \(\frac{\delta y}{\delta x} = 2\)
\(\delta x = \frac{\delta u}{2}\)
Now \(\int (2x + 3)^{\frac{1}{2}} \delta x = \int u^{\frac{1}{2}}.{\frac{\delta x}{2}}\)
\( = \frac{1}{2} \int u^{\frac{1}{2}} \delta u\)
\( = \frac{1}{2} u^{\frac{3}{2}} \times \frac{2}{3} + k\)
\( = \frac{1}{3} u^{\frac{3}{2}} + k\)
\( = \frac{1}{3} (2x + 3)^{\frac{3}{2}} + k\)
let u = 2x + 3, \(\frac{\delta y}{\delta x} = 2\)
\(\delta x = \frac{\delta u}{2}\)
Now \(\int (2x + 3)^{\frac{1}{2}} \delta x = \int u^{\frac{1}{2}}.{\frac{\delta x}{2}}\)
\( = \frac{1}{2} \int u^{\frac{1}{2}} \delta u\)
\( = \frac{1}{2} u^{\frac{3}{2}} \times \frac{2}{3} + k\)
\( = \frac{1}{3} u^{\frac{3}{2}} + k\)
\( = \frac{1}{3} (2x + 3)^{\frac{3}{2}} + k\)
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