8.5.1 Integration by substitution Quiz: Pearson Edexcel Maths, Unit 8

The 20 questions

1. What is the primary purpose of integration by substitution in calculus?

   • Calculate limits
   • Simplify integrals
   • Find derivatives
   • Factorise polynomials

2. When using substitution $u = g(x)$, what must be replaced alongside the function?

   • The power
   • The limit
   • The constant
   • dx

3. If $u = 3x + 1$, what is the correct expression for du in terms of dx?

   • 3 dx
   • x dx
   • dx
   • 1/3 dx

4. For definite integrals, what must also change when applying a substitution?

   • Coefficient
   • Integrand
   • Limits
   • Variable

5. Integration by substitution is the reverse process of which differentiation rule?

   • Power rule
   • Quotient rule
   • Chain rule
   • Product rule

6. What substitution is best suited for integrating x multiplied by a function of x squared?

   • u = root x
   • u = x squared
   • u = x
   • u = x cubed

7. Evaluate the integral of 2x multiplied by e to the power of x squared.

   • 2x e to the x squared
   • e to the 2x
   • e to the x squared
   • 2 e to the x

8. What is the integral of f prime of x divided by f of x?

   • ln mod f of x
   • 1 over f of x
   • f of x squared
   • e to the f of x

9. What is the integral of 2x/(x^2 + 1) with respect to x?

   • 1/(x^2 + 1)
   • ln|x^2 + 1|
   • arctan(x)
   • 2 ln|x^2 + 1|

10. Which algebraic technique often precedes substitution when integrating rational functions?

    • Partial fractions
    • Polynomial long division
    • Rationalising the denominator
    • Completing the square

11. What substitution should be used to integrate x times the square root of x plus 1?

    • u = x plus 1
    • u = x squared
    • u = root x
    • u = x

12. If u = x squared minus 3, what is du for an integration problem?

    • 2 dx
    • x squared dx
    • x dx
    • 2x dx

13. When transforming definite integral limits, if lower limit x = 0 and u = 2x + 1, new lower limit is...

    • 2
    • 1
    • 0
    • 1 over 2

14. What is the integral of tan x with respect to x using standard substitution methods?

    • -ln mod cos x
    • ln mod sec x
    • ln mod sin x
    • sec squared x

15. Which trigonometric substitution is typically used for integrals containing a squared minus x squared?

    • x = a sin theta
    • x = a cosh theta
    • x = a tan theta
    • x = a sec theta

16. Evaluate the integral of cos(x) sin^3(x) with respect to x.

    • 1/3 sin^3(x)
    • 4 sin^4(x)
    • 1/4 sin^4(x)
    • sin^4(x)

17. If u = ln x, what is the differential du?

    • e to the x dx
    • 1 over ln x dx
    • x dx
    • 1 over x dx

18. Find the integral of 1 over x ln x with respect to x.

    • 1 over x squared
    • ln mod ln x
    • 1 over ln x
    • ln x

19. What substitution transforms the integral of 1 over 1 plus x squared into standard form?

    • x = sin theta
    • x = sec theta
    • x = tan theta
    • x = sinh theta

20. When integrating by substitution, failure to change what component usually leads to incorrect definite values?

    • Limits
    • Integrand
    • Exponent
    • Coefficient

Related quizzes

• Fundamental Theorem of calculus and integration as reverse differentiation Quiz · 8.1.1 · 20 questions
• Integrating x^n and related sums Quiz · 8.2.1 · 20 questions
• Definite integrals and areas Quiz · 8.3.1 · 20 questions
• Integrating exponentials and reciprocal functions Quiz · 8.4.1 · 20 questions
• Integration using partial fractions Quiz · 8.6.1 · 20 questions
• Solving first order differential equations analytically Quiz · 8.7.1 · 20 questions
• Interpreting solutions of differential equations Quiz · 8.8.1 · 20 questions
• Structure of mathematical proof and methods of proof Quiz · 1.1.1 · 20 questions
• Vectors in two and three dimensions Quiz · 10.1.1 · 20 questions
• Laws of indices for all rational exponents Quiz · 2.1.1 · 20 questions