≤ft(7x²-3x+8-x^{-1/2}+x^{-1}+x^{-2})\,dx

3/7x³-2/3x²+8x-1/2x¹/2}+ x+x^{-1}

7/3x³+3/2x²+8x-2x¹/2}+ x+x^{-1}

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Integral Calculus

Question : 45 of 73

Marks: +1, -0

\int\left(7x^{2}-3x+8-x^{-1/2}+x^{-1}+x^{-2}\right)\,dx

Solution:

Concept:

Integrate each term using the power rule

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

for

n \neq -1

, and

\int x^{-1} dx = \ln|x|

Explanation:

Compute each integral separately:

\int 7x^2 dx = 7 \cdot \frac{x^3}{3} = \frac{7}{3}x^3

\int -3x dx = -3 \cdot \frac{x^2}{2} = -\frac{3}{2}x^2

\int 8 dx = 8x

\int -x^{-1/2} dx = - \frac{x^{1/2}}{1/2} = -2x^{1/2}

\int x^{-1} dx = \ln|x|

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

Combine all terms and add the constant of integration

C

\frac{7}{3}x^3 - \frac{3}{2}x^2 + 8x - 2x^{1/2} + \ln|x| - x^{-1} + C

Compare with options:

Option A:

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

— does not match.

Option B:

\frac{3}{7}x^3 - \frac{2}{3}x^2 + 8x - \frac{1}{2}x^{1/2} + \log x + x^{-1}

— coefficients and signs wrong.

Option C:

\frac{7}{3}x^3 + \frac{3}{2}x^2 + 8x - 2x^{1/2} + \log x + x^{-1}

— signs wrong on

x^2

and

x^{-1}

terms.

Option D: None — correct.

Answer:

None

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