The anti-derivative of - x -1- x equalsb frac 23 x -lfrac23- frac 12 x - 2- Cllc frac23 x -frac32-2 x -d frac 32 x - frac 32- frac 12 x - frac -12- Cl

∫ ( √(x)+1/√(x) )dx =∫√(x)dx +∫1/√(x)dx =∫ x^1/2dx +∫ x^ 1/2dx =x^3/2/3/2+x^1/2/1/2+C =2/3x^3/2+2x^1/2+C Hence, the correct option is (C). ...

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Standard XII

Mathematics

First Fundamental Theorem of Calculus

The anti-deri...

Question

The anti-derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
$(a) \frac{1}{3} x^{\frac{1}{3}} + 2 x^{\frac{1}{2}} + C (b) \frac{2}{3} x^{\frac{2}{3}} + \frac{1}{2} x^{2} + C (c) \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C (d) \frac{3}{2} x^{\frac{3}{2}} + \frac{1}{2} x^{\frac{1}{2}} + C$

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Solution

$\int (\sqrt{x} + \frac{1}{\sqrt{x}}) d x = \int \sqrt{x} d x + \int \frac{1}{\sqrt{x}} d x = \int x^{\frac{1}{2}} d x + \int x^{- \frac{1}{2}} d x = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C$
Hence, the correct option is (C).

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The anti-derivative of (√x+1√x) equals (a)13x13+2x12+C(b)23x23+12x2+C(c)23x32+2x12+C(d)32x32+12x12+C

∫(√x+1√x)dx=∫√xdx+∫1√xdx=∫x12dx+∫x−12dx=x3232+x1212+C=23x32+2x12+C Hence, the correct option is (C).

The anti-derivative of $(\sqrt{x} + \frac{1}{\sqrt{x}})$ equals
$(a) \frac{1}{3} x^{\frac{1}{3}} + 2 x^{\frac{1}{2}} + C (b) \frac{2}{3} x^{\frac{2}{3}} + \frac{1}{2} x^{2} + C (c) \frac{2}{3} x^{\frac{3}{2}} + 2 x^{\frac{1}{2}} + C (d) \frac{3}{2} x^{\frac{3}{2}} + \frac{1}{2} x^{\frac{1}{2}} + C$