ARCH-MM-10: Definite Integration of Rational Powers (x^n, n ≠ -1) & Area Enclosed Between Curves

Full 15-Q Timed Drill

Step 1 of 6 • The Core Idea

ARCH-MM-10: Why the 180s Classroom Method Fails Under the 90s Clock

Standard Classroom Method (Too Slow)~180s (Blows 90s Clock)

Rewriting

\frac{2x^2 - 3}{x^{3/2}} = 2x^{1/2} - 3x^{-3/2}

and integrating gives

\left[\frac{4}{3}x^{3/2} + 6x^{-1/2}\right]_1^4 = \left(\frac{32}{3} + 3\right) - \left(\frac{4}{3} + 6\right) = \frac{19}{3}

(~135s).

The Fast ShortcutTarget: 50s (Saves 130s)

Line 1 (Split & Integrate Powers):

\int_1^4 (2x^{1/2} - 3x^{-3/2})\,dx = \left[\frac{4}{3}x^{3/2} + \frac{6}{\sqrt{x}}\right]_1^4

.
Line 2 (Bracket Differences):

\frac{4}{3}(8 - 1) + 6\left(\frac{1}{2} - 1\right) = \frac{28}{3} - 3 = \frac{19}{3}

(35s).

Where Students Lose Time: In 22 past UAT-UK questions (5.37% of Mathematics 2), candidates lose 130s by confusing signed definite integral with geometric area |int_a^b f| + |int_b^c f|. Instead, apply MM7 with Maths 1 skill M4.14.

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