#8747;abfxdx=limh #8594;0 #160;hfa+fa+h+fa+2h...............+fa+n 1hwhere #160;h=b an Here, #160;a=1, #160;b=3, #160;fx=2x2+5x, #160;h=3 1n=2n ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Harmonic Progression

∫ 132 x 2+5 x...

Question

$\int_{1}^{3} (2 x^{2} + 5 x) d x$

Open in App

Solution

$\int_{a}^{b} f (x) d x = \lim_{h \to 0} h [f (a) + f (a + h) + f (a + 2 h) . . . . . . . . . . . . . . . + f \{a + (n - 1) h\}] where h = \frac{b - a}{n}$

$Here, a = 1, b = 3, f (x) = 2 x^{2} + 5 x, h = \frac{3 - 1}{n} = \frac{2}{n} Therefore, I = \int_{1}^{3} (2 x^{2} + 5 x) d x = \lim_{h \to 0} h [f (1) + f (1 + h) + . . . . . . . . . . . . . . . . . . . . + f \{1 + (n - 1) h\}] = \lim_{h \to 0} h [(2 + 5) + \{2 {(1 + h)}^{2} + 5 (1 + h)\} + . . . . . . . . . . . . . . . + \{2 {(1 + (n - 1) h)}^{2} + 5 (1 + (n - 1) h)\}] = \lim_{h \to 0} h [2 \{1^{2} + {(1 + h)}^{2} + . . . . . . . . . . . . + {\{1 + (n - 1) h\}}^{2}\} + 5 \{1 + (1 + h) + (1 + 2 h + . . . . . . . . + (1 + (n - 1) h))\}] = \lim_{h \to 0} h [2 n + 2 h^{2} (1^{2} + 2^{2} + 3^{2} . . . . . . . . . + {(n - 1)}^{2}) + 4 h \{1 + 2 + . . . . . . + (n - 1)\} + 5 n + 5 h \{1 + 2 + . . . . . . + (n - 1)\}] = \lim_{h \to 0} h [7 n + 2 h^{2} \frac{n (n - 1) (2 n - 1)}{6} + 9 h \frac{n (n - 1)}{2}] = \lim_{n \to \infty} \frac{2}{n} [7 n + \frac{4 (n - 1) (2 n - 1)}{3 n} + 9 n - 9] = \lim_{n \to \infty} 2 [16 + \frac{4}{3} (1 - \frac{1}{n}) (2 - \frac{1}{n}) - \frac{9}{n}] = 32 + \frac{16}{3} = \frac{112}{3}$

Suggest Corrections