Hard A-Level Standard integrals Questions

Challenging, exam-style A-Level Standard integrals questions with worked solutions. Stretch yourself on the hardest standard integrals, exponential, trigonometric, definite integral problems.

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A-Level34 questionsStep-by-step solutions
Question 1
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
Question 2
8 markschallenging
Which expression correctly gives ∫1x dx\int \frac{1}{x}\,dx?
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫1x dx\int \frac{1}{x}\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=1x\text{integrand} = \frac{1}{x}

    Recognise it as a reciprocal function so we know which rule to use.

  3. Recall the relevant standard result

    ∫1x dx=ln⁡∣x∣+c\int \frac{1}{x}\,dx=\ln\left|x\right|+c

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

  4. Apply the standard result

    ∫1x dx=ln⁡∣x∣\int \frac{1}{x}\,dx = \ln\left|x\right|

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(ln⁡∣x∣)=1x\frac{d}{dx}\left(\ln\left|x\right|\right)=\frac{1}{x}

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    ln⁡∣x∣+c\ln\left|x\right|+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    −1x2+c does not differentiate to 1x-\frac{1}{x^{2}}+c\ \text{does not differentiate to }\frac{1}{x}

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

  8. Eliminate the second incorrect option

    1x2+c does not differentiate to 1x\frac{1}{x^{2}}+c\ \text{does not differentiate to }\frac{1}{x}

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

  9. Eliminate the third incorrect option

    ln⁡(x)+c does not differentiate to 1x\ln\left(x\right)+c\ \text{does not differentiate to }\frac{1}{x}

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

  10. Eliminate the fourth incorrect option

    xln⁡∣x∣−x+c does not differentiate to 1xx\ln\left|x\right|-x+c\ \text{does not differentiate to }\frac{1}{x}

    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

    ∫1x dx=ln⁡∣x∣+c\int \frac{1}{x}\,dx=\ln\left|x\right|+c

    The same rule applies to every function of this type.

  14. Confirm the coefficient in the answer

    ln⁡∣x∣\ln\left|x\right|

    The coefficient comes directly from the standard result.

  15. Select the correct option

    ln⁡∣x∣+c\ln\left|x\right|+c

    This is the only expression whose derivative is the integrand.

Answer
ln⁡∣x∣+c\ln\left|x\right|+c
Question 3
8 markschallenging
Select the correct expression for ∫sec⁡2(2x) dx\int \sec^{2}\left(2 x\right)\,dx.
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫sec⁡2(2x) dx\int \sec^{2}\left(2 x\right)\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=sec⁡2(2x)\text{integrand} = \sec^{2}\left(2 x\right)

    Recognise it as a squared secant function so we know which rule to use.

  3. Recall the relevant standard result

    ∫sec⁡2(kx) dx=1ktan⁡(kx)+c\int \sec^{2}(kx)\,dx=\frac{1}{k}\tan(kx)+c

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

  4. Apply the standard result

    ∫sec⁡2(2x) dx=12tan⁡(2x)\int \sec^{2}\left(2 x\right)\,dx = \frac{1}{2}\tan\left(2 x\right)

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(12tan⁡(2x))=sec⁡2(2x)\frac{d}{dx}\left(\frac{1}{2}\tan\left(2 x\right)\right)=\sec^{2}\left(2 x\right)

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    12tan⁡(2x)+c\frac{1}{2}\tan\left(2 x\right)+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    2tan⁡(2x)+c does not differentiate to sec⁡2(2x)2\tan\left(2 x\right)+c\ \text{does not differentiate to }\sec^{2}\left(2 x\right)

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

  8. Eliminate the second incorrect option

    12sec⁡(2x)+c does not differentiate to sec⁡2(2x)\frac{1}{2}\sec\left(2 x\right)+c\ \text{does not differentiate to }\sec^{2}\left(2 x\right)

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

  9. Eliminate the third incorrect option

    tan⁡(2x)+c does not differentiate to sec⁡2(2x)\tan\left(2 x\right)+c\ \text{does not differentiate to }\sec^{2}\left(2 x\right)

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

  10. Eliminate the fourth incorrect option

    −12tan⁡(2x)+c does not differentiate to sec⁡2(2x)-\frac{1}{2}\tan\left(2 x\right)+c\ \text{does not differentiate to }\sec^{2}\left(2 x\right)

    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

    ∫sec⁡2(kx) dx=1ktan⁡(kx)+c\int \sec^{2}(kx)\,dx=\frac{1}{k}\tan(kx)+c

    The same rule applies to every function of this type.

  14. Confirm the coefficient in the answer

    12tan⁡(2x)\frac{1}{2}\tan\left(2 x\right)

    The coefficient comes directly from the standard result.

  15. Select the correct option

    12tan⁡(2x)+c\frac{1}{2}\tan\left(2 x\right)+c

    This is the only expression whose derivative is the integrand.

Answer
12tan⁡(2x)+c\frac{1}{2}\tan\left(2 x\right)+c
Question 4
8 markschallenging
Which of the following is ∫cos⁡(3x) dx\int \cos\left(3 x\right)\,dx?
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫cos⁡(3x) dx\int \cos\left(3 x\right)\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=cos⁡(3x)\text{integrand} = \cos\left(3 x\right)

    Recognise it as a cosine function so we know which rule to use.

  3. Recall the relevant standard result

    ∫cos⁡(kx) dx=1ksin⁡(kx)+c\int \cos(kx)\,dx=\frac{1}{k}\sin(kx)+c

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

  4. Apply the standard result

    ∫cos⁡(3x) dx=13sin⁡(3x)\int \cos\left(3 x\right)\,dx = \frac{1}{3}\sin\left(3 x\right)

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(13sin⁡(3x))=cos⁡(3x)\frac{d}{dx}\left(\frac{1}{3}\sin\left(3 x\right)\right)=\cos\left(3 x\right)

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    13sin⁡(3x)+c\frac{1}{3}\sin\left(3 x\right)+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    −13sin⁡(3x)+c does not differentiate to cos⁡(3x)-\frac{1}{3}\sin\left(3 x\right)+c\ \text{does not differentiate to }\cos\left(3 x\right)

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

  8. Eliminate the second incorrect option

    3sin⁡(3x)+c does not differentiate to cos⁡(3x)3\sin\left(3 x\right)+c\ \text{does not differentiate to }\cos\left(3 x\right)

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

  9. Eliminate the third incorrect option

    −13cos⁡(3x)+c does not differentiate to cos⁡(3x)-\frac{1}{3}\cos\left(3 x\right)+c\ \text{does not differentiate to }\cos\left(3 x\right)

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

  10. Eliminate the fourth incorrect option

    13cos⁡(3x)+c does not differentiate to cos⁡(3x)\frac{1}{3}\cos\left(3 x\right)+c\ \text{does not differentiate to }\cos\left(3 x\right)

    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

    ∫cos⁡(kx) dx=1ksin⁡(kx)+c\int \cos(kx)\,dx=\frac{1}{k}\sin(kx)+c

    The same rule applies to every function of this type.

  14. Confirm the coefficient in the answer

    13sin⁡(3x)\frac{1}{3}\sin\left(3 x\right)

    The coefficient comes directly from the standard result.

  15. Select the correct option

    13sin⁡(3x)+c\frac{1}{3}\sin\left(3 x\right)+c

    This is the only expression whose derivative is the integrand.

Answer
13sin⁡(3x)+c\frac{1}{3}\sin\left(3 x\right)+c
Question 5
8 markschallenging
Using the standard integration results, determine ∫sin⁡(2x) dx\int \sin\left(2 x\right)\,dx.
Show worked solution

Worked solution

  1. Write down the integral to evaluate

    ∫sin⁡(2x) dx\int \sin\left(2 x\right)\,dx

    Identify exactly what must be integrated.

  2. Identify the type of function

    integrand=sin⁡(2x)\text{integrand} = \sin\left(2 x\right)

    Recognise it as a sine function so we know which rule to use.

  3. Recall the relevant standard result

    ∫sin⁡(kx) dx=−1kcos⁡(kx)+c\int \sin(kx)\,dx=-\frac{1}{k}\cos(kx)+c

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

  4. Apply the standard result

    ∫sin⁡(2x) dx=−12cos⁡(2x)\int \sin\left(2 x\right)\,dx = -\frac{1}{2}\cos\left(2 x\right)

    Substitute into the standard result.

  5. Check by differentiating the result

    ddx(−12cos⁡(2x))=sin⁡(2x)\frac{d}{dx}\left(-\frac{1}{2}\cos\left(2 x\right)\right)=\sin\left(2 x\right)

    Differentiating the proposed answer returns the integrand.

  6. Include the constant of integration

    −12cos⁡(2x)+c-\frac{1}{2}\cos\left(2 x\right)+c

    Indefinite integrals require the arbitrary constant c.

  7. Eliminate the first incorrect option

    12cos⁡(2x)+c does not differentiate to sin⁡(2x)\frac{1}{2}\cos\left(2 x\right)+c\ \text{does not differentiate to }\sin\left(2 x\right)

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

  8. Eliminate the second incorrect option

    −2cos⁡(2x)+c does not differentiate to sin⁡(2x)-2\cos\left(2 x\right)+c\ \text{does not differentiate to }\sin\left(2 x\right)

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

  9. Eliminate the third incorrect option

    −12sin⁡(2x)+c does not differentiate to sin⁡(2x)-\frac{1}{2}\sin\left(2 x\right)+c\ \text{does not differentiate to }\sin\left(2 x\right)

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

  10. Eliminate the fourth incorrect option

    12sin⁡(2x)+c does not differentiate to sin⁡(2x)\frac{1}{2}\sin\left(2 x\right)+c\ \text{does not differentiate to }\sin\left(2 x\right)

    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

    ∫sin⁡(kx) dx=−1kcos⁡(kx)+c\int \sin(kx)\,dx=-\frac{1}{k}\cos(kx)+c

    The same rule applies to every function of this type.

  14. Confirm the coefficient in the answer

    −12cos⁡(2x)-\frac{1}{2}\cos\left(2 x\right)

    The coefficient comes directly from the standard result.

  15. Select the correct option

    −12cos⁡(2x)+c-\frac{1}{2}\cos\left(2 x\right)+c

    This is the only expression whose derivative is the integrand.

Answer
−12cos⁡(2x)+c-\frac{1}{2}\cos\left(2 x\right)+c

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