Mathematics for Management -- Supplementary Electronic Materials

Quiz: The Substitution Rule for Definite Integrals

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1. Determine the value of \( \int^1_0 \left( 2x - 1 \right)^3 \textrm{d} x\) .

\(\frac{1}{4} \) \( \frac{1}{2} \) \( 0 \) \( 2 \)

2. Determine the value of \( \int^{\pi}_0 \, \cos^2(x) \sin(x) \, \textrm{d} x\) .

\( -\frac{2}{3} \) \( \frac{1}{3} \) \( 1 \) \( \frac{2}{3} \)

3. Determine the value of \( \int^{\rm{e}}_1 \frac{\ln(x)}{x} \textrm{d} x\) .

\( \frac{1}{2}( {\rm{e}}^2 - 1 ) \) \( \frac{1}{2} \) \( {\rm{e}} - 1 \) \( 1 \)

4. Determine the value of \( \int^{\pi/2}_{\pi/4} \frac{\cos(x)}{\sin(x)} \textrm{d} x \).

\( \ln(\sqrt{2}) \) \( \ln(\frac{\pi}{4}) \) \( \ln(\sqrt{3}) \) \( \ln(\frac{\sqrt{3}}{2}) \)

5. Determine the value of \( \int^{1/2}_0 \frac{2x}{\sqrt{1-x^2}} \textrm{d} x \).

\( 1 - \sqrt{3}{2} \) \( \frac{\pi}{6} \) \( \frac{\pi}{6} - 1 \) \( 2 - \sqrt{3} \)

6. Determine the value of \( \int^1_0 \left( x+1 \right) {\rm{e}}^{x^2 + 2x} \textrm{d} x \).

\( \frac{1}{2} {\rm{e}}^3 \) \( \frac{1}{2} \left( {\rm{e}}^3 - 1 \right) \) \( {\rm{e}}^3 - 1 \) \( \frac{1}{2} \left( 1 - {\rm{e}}^3 \right) \)

7. Determine the value of \( \int^2_1 \frac{x+1}{x^2 + 2x} \textrm{d} x \).

\( \ln(8) - \ln(3) \) \( \frac{1}{2} \left( \ln(8) - \ln(3) \right) \) \( \frac{3}{2} \ln(2) \) \( \frac{1}{2} \left( 3 \ln(2) + 2 \right) \)

8. If the substitution \( u = \frac{1}{2} x \) is applied in the integrand of \( \int^4_2 \frac{1 - \frac{1}{4} x^2}{x} \textrm{d} x \), the resulting integral is

\( \int^2_1 \frac{1 - u^2}{u} \textrm{d} u \) \( \int^2_1 \frac{1 - u^2}{2u} \textrm{d} u \) \( \int^4_2 \frac{1 - u^2}{u} \textrm{d} u \) \( \int^2_1 \frac{1 - u^2}{4u} \textrm{d} u \)

9. If the substitution \( \sqrt{x} = \sin(y) \) is applied in the integrand of \( \int^{1/2}_0 \frac{\sqrt{x}}{\sqrt{1-x}} \textrm{d} x \), the resulting integral is

\( \int^{1/2}_{0} \, \sin^2(y) \, \textrm{d} y \) \( 2 \int^{\pi/4}_{0} \, \sin^2(y) \, \textrm{d} y \) \( \int^{\pi/4}_{0} \, \sin^2(y) \, \textrm{d} y \) \( 2 \int^{\pi/6}_{0} \, \sin^2(y) \, \textrm{d} y \)

10. If \( \int^2_1 \, f(x-c) \, \textrm{d} x = 5 \), where \( c \) is a real constant, then which value will \( \int^{2-c}_{1-c} \, f(x) \, \textrm{d} x \) have?

\( 5+c \) \( 5 \) \( 5-c \) \( c-5 \)

Your grade is: __ of 10

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

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