Mathematics for Management -- Supplementary Electronic Materials

Indefinite integration

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1. Let \(f(x) = x^{-9}\). Determine the general equation for the antiderivative of \(f(x)\).

\( \frac{1}{8} x^{-8} + C \) \( \frac{1}{10} x^{-10} + C \) \( -\frac{1}{8} x^{-8} + C \) \( -\frac{1}{10} x^{-10} + C \)

2. Let \(f(x) = 4x-7\). Determine the general equation for the antiderivative of \(f(x)\).

\( 4 + C \) \( \frac{1}{2} x^2 - 7x + C \) \( 2x^2 - 7x + C \) \( 8x^2 + C \)

3. Let \(f(x) = 6 {\rm{e}}^x\). Determine the general equation for the antiderivative of \(f(x)\).

\( \frac{6}{x+1} {\rm{e}}^{x+1} + C \) \( 6 x {\rm{e}}^{x-1} + C \) \( 6 \ln|x| + C \) \( 6 {\rm{e}}^x + C \)

4. Let \(f(x) = {\rm{e}}^{kx}\). Determine the indefinite integral of \(f(x)\).

\( \frac{1}{k} {\rm{e}}^{kx} + C \) \( {\rm{e}}^x + C \) \( -{\rm{e}}^x + C \) \( \frac{1}{k} \sin(kx) + C \)

5. Determine \(\int \, \cos(3x) \textrm{d} x\)

\( -3 \sin(3x) + C \) \( -\tfrac{1}{3} \sin(3x) + C \) \( 3 \sin(3x) + C \) \( \tfrac{1}{3} \sin(3x) + C \)

6. Determine \(\int \left( x^3 - 3x \right) \textrm{d} x\).

\( 3x^2 - 3 + C \) \( \frac{x^4}{4} - \frac{3x^2}{2} + C \) \( \frac{x^4}{4} - 3x^2 + C \) \( 4x^4 - 6x^2 + C \)

7. Determine \(\int \left( x^3 + 1 \right)^2 \textrm{d} x\).

\( \frac{1}{7} x^7 +x + C \) \( \frac{1}{7} x^7 + \frac{1}{2} x^4 + x + C \) \( 6 x^2 \left( x^3 + 1 \right) + C \) \( \frac{\left( x^3 + 1 \right)^3}{9 x^2} + C \)

8. Determine \(\int \frac{x^2}{{\rm{e}}^{x^3}} \textrm{d} x\).

\( -\frac{1}{3} \ln( {\rm{e}}^{x^3} ) + C \) \( -\frac{{\rm{e}}^{x^3}}{3} + C \) \( -\frac{1}{3 {\rm{e}}^{x^3}} + C \) \( \frac{x^3}{3 {\rm{e}}^{x^3}} + C \)

9. Determine \(\int \left( \sqrt{x} + \cos(x) \right) \textrm{d} x\).

\( \frac{2}{3} x^{3/2} + \sin(x) \) \( \frac{1}{2} x^{-1/2} - \sin(x) \) \( \frac{1}{2 \sqrt{x}} + \sin(x) \) \( \sqrt{\frac{x^2}{2}} + \cos\left( \frac{x^2}{2} \right) \)

10. Determine \(\int \left( \frac{x^3 - x - 1}{x^2} \right) \textrm{d} x\).

\( \frac{\frac{1}{4} x^4 - \frac{1}{2} x^2 - x}{\frac{1}{3} x^3} + C \) \( 1 + \frac{1}{x^2} + \frac{2}{x^3} + C \) \( \frac{1}{2} x^2 - \ln|x| + \frac{1}{x} + C \) \( \frac{1}{2} x^2 - \ln|x| - \frac{1}{x} + C \)

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Solution: 1c; 2c; 3d; 4a; 5d; 6b; 7b; 8c; 9a; 10c

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.