Integration of {x} - 1/2 x + {2}{{x}} is

Correct Answer is : 32x32−x2+x12+c3/2 x³/2} - x² + x¹/2} + c23x23−x2+x21+c

2/3 x³/2} - 1/4 x² + 4 x¹/2} + c

2/3 x³/2} - x² + 2 x¹/2} + c

Correct Answer is : 32x32−x2+x12+c3/2 x³/2} - x² + x¹/2} + c23x23−x2+x21+c

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Integral Calculus

Question : 1 of 73

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Integration of

\sqrt{x} - \frac{1}{2} x + \frac{2}{\sqrt{x}}

Solution:

Concept:

Integrate each term separately using the power rule

\int x^n \, dx = \frac{x^{n+1}}{n+1} + c

Explanation:

\int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}

\int -\frac{1}{2}x \, dx = -\frac{1}{2} \cdot \frac{x^2}{2} = -\frac{1}{4} x^2

\int \frac{2}{\sqrt{x}} \, dx = 2 \int x^{-1/2} \, dx = 2 \cdot \frac{x^{1/2}}{1/2} = 4 x^{1/2}

Adding all terms and the constant of integration

c

gives

\frac{2}{3} x^{3/2} - \frac{1}{4} x^2 + 4 x^{1/2} + c

Answer:

Option A:

\frac{2}{3} x^{\frac{3}{2}} - \frac{1}{4} x^{2} + 4 x^{\frac{1}{2}} + c

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