A-Level Standard integrals Practice Questions

Free A-Level Standard integrals practice questions with full step-by-step worked solutions. Covers standard integrals, power rule, exponential, trigonometric. Practise exam-style problems and check your method.

standard integralspower ruleexponentialtrigonometriclogarithmicdefinite integral
A-Level70 questionsStep-by-step solutions
Question 1
2 markseasy
Find ∫x3 dx\int x^{3}\,dx.
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Worked solution

  1. Write down the integral

    ∫x3 dx\int x^{3}\,dx

    We need a function whose derivative is the integrand.

  2. Apply the standard result

    ∫x3 dx=14x4\int x^{3}\,dx = \frac{1}{4}x^{4}

    Use the formula-book result for this type of term.

  3. Add the constant of integration

    14x4+c\frac{1}{4}x^{4} + c

    An indefinite integral always includes an arbitrary constant c.

Answer
14x4+c\frac{1}{4}x^{4} + c
Question 2
2 markseasy
Which of the following is ∫x4 dx\int x^{4}\,dx?
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Worked solution

  1. Recall the relevant standard result

    ∫xn dx=xn+1n+1+c (n≠−1)\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1)

    Use the formula-book result for this type of function.

  2. Apply the result to the given function

    ∫x4 dx=x55+c\int x^{4}\,dx = \frac{x^{5}}{5}+c

    Integrate, remembering the constant of integration.

  3. Select the correct option

    x55+c\frac{x^{5}}{5}+c

    This is the expression whose derivative is the integrand.

Answer
x55+c\frac{x^{5}}{5}+c
Question 3
3 marksintermediate
Which of the following is ∫x5 dx\int x^{5}\,dx?
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Worked solution

  1. Write down the integral to evaluate

    ∫x5 dx\int x^{5}\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=x5\text{integrand} = x^{5}

    Recognise it as a power of x so we know which rule to use.

  3. Recall the relevant standard result

    ∫xn dx=xn+1n+1+c (n≠−1)\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1)

    This is the formula-book result for this type of function.

  4. Apply the standard result

    ∫x5 dx=x66\int x^{5}\,dx = \frac{x^{6}}{6}

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(x66)=x5\frac{d}{dx}\left(\frac{x^{6}}{6}\right)=x^{5}

    Differentiating the proposed answer returns the integrand.

  6. Select the correct option

    x66+c\frac{x^{6}}{6}+c

    This is the only expression whose derivative is the integrand.

Answer
x66+c\frac{x^{6}}{6}+c
Question 4
5 markshard
Which of the following is ∫x6 dx\int x^{6}\,dx?
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫x6 dx\int x^{6}\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=x6\text{integrand} = x^{6}

    Recognise it as a power of x so we know which rule to use.

  3. Recall the relevant standard result

    ∫xn dx=xn+1n+1+c (n≠−1)\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1)

    This is the formula-book result for this type of function.

  4. Apply the standard result

    ∫x6 dx=x77\int x^{6}\,dx = \frac{x^{7}}{7}

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(x77)=x6\frac{d}{dx}\left(\frac{x^{7}}{7}\right)=x^{6}

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    x77+c\frac{x^{7}}{7}+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    x7+c does not differentiate to x6x^{7}+c\ \text{does not differentiate to }x^{6}

    Its derivative is not the integrand, so it is rejected.

  8. Eliminate the second incorrect option

    x76+c does not differentiate to x6\frac{x^{7}}{6}+c\ \text{does not differentiate to }x^{6}

    Its derivative is not the integrand, so it is rejected.

  9. Eliminate the third incorrect option

    6x5+c does not differentiate to x66x^{5}+c\ \text{does not differentiate to }x^{6}

    Its derivative is not the integrand, so it is rejected.

  10. Select the correct option

    x77+c\frac{x^{7}}{7}+c

    This is the only expression whose derivative is the integrand.

Answer
x77+c\frac{x^{7}}{7}+c
Question 5
8 markschallenging
Which of the following is ∫x7 dx\int x^{7}\,dx?
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫x7 dx\int x^{7}\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=x7\text{integrand} = x^{7}

    Recognise it as a power of x so we know which rule to use.

  3. Recall the relevant standard result

    ∫xn dx=xn+1n+1+c (n≠−1)\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1)

    This is the formula-book result for this type of function.

  4. Apply the standard result

    ∫x7 dx=x88\int x^{7}\,dx = \frac{x^{8}}{8}

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(x88)=x7\frac{d}{dx}\left(\frac{x^{8}}{8}\right)=x^{7}

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    x88+c\frac{x^{8}}{8}+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    x8+c does not differentiate to x7x^{8}+c\ \text{does not differentiate to }x^{7}

    Its derivative is not the integrand, so it is rejected.

  8. Eliminate the second incorrect option

    x87+c does not differentiate to x7\frac{x^{8}}{7}+c\ \text{does not differentiate to }x^{7}

    Its derivative is not the integrand, so it is rejected.

  9. Eliminate the third incorrect option

    7x6+c does not differentiate to x77x^{6}+c\ \text{does not differentiate to }x^{7}

    Its derivative is not the integrand, so it is rejected.

  10. Eliminate the fourth incorrect option

    8x7+c does not differentiate to x78x^{7}+c\ \text{does not differentiate to }x^{7}

    Its derivative is not the integrand, so it is rejected.

  11. Note the common coefficient error

    divide by the coefficient; do not multiply\text{divide by the coefficient; do not multiply}

    A frequent mistake is multiplying by the constant instead of dividing.

  12. Note the common sign error

    keep the sign given by the standard result\text{keep the sign given by the standard result}

    Sine and cosine integrals change sign, so check carefully.

  13. Restate the standard result in general form

    ∫xn dx=xn+1n+1+c (n≠−1)\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+c\ (n\neq-1)

    The same rule applies to every function of this type.

  14. Confirm the coefficient in the answer

    x88\frac{x^{8}}{8}

    The coefficient comes directly from the standard result.

  15. Select the correct option

    x88+c\frac{x^{8}}{8}+c

    This is the only expression whose derivative is the integrand.

Answer
x88+c\frac{x^{8}}{8}+c

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