Substitution works when a change of variable transforms the entire integrand, including dx. Look for a function paired with its derivative. Then differentiate your answer: this is a stronger check than comparing the appearance of two expressions.
Curriculum checked 2 October 2026 · original practice for selected skills.
You will need: Chain rule, powers and natural logarithms.
Attempt each task before opening a hint. If you need help, reveal the first hint, then the setup, then compare your written steps with the solution. Finish with the retests without referring back.
If u is a function of x, du includes its derivative. Replacing the inner expression alone leaves the substitution incomplete.
An indefinite integral represents a family of antiderivatives, so include an arbitrary constant C.
For an integrand of the form f′(x)/f(x), a logarithm often appears. Use the absolute value where needed on an interval where f(x) is non-zero.
The derivative of x²+1 is already present.
Let u=x²+1, so du=2x dx.
The complete integral becomes a power of u.
Replace u; differentiating brings back the factor 2x.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
Write a method as well as an answer. The tasks change the reasoning, not just the numbers.
The inner derivative is twice the x you have.
Let u=x²+4; x dx=du/2.
Transform the differential as well as the bracket.
Replace u by x²+4.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
Compare the numerator with the denominator's derivative.
u=3x+2 gives du=3 dx.
The integral is 1/u with respect to u.
Use an absolute value for the real logarithm.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
Differentiation would produce a factor of 4.
Compensate by multiplying the antiderivative by 1/4.
u=4x gives dx=du/4.
Differentiating the result cancels the quarter.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
A tempting claim: “Replace the bracket by u and leave dx unchanged.”
The differential changes at a rate determined by the inner function.
These changed questions test whether you can reconstruct the method. If you use a hint, record where you got stuck and retry later. Completing this small set does not establish full chapter mastery.
Express 6x dx in terms of du.
For u=x²+2, 6x dx=3du.
Obtain 3 times the integral of u².
The factor 3 cancels the power-rule denominator.
Answer:
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
The derivative of the denominator is 5.
dx=du/5 when u=5x−1.
Keep the factor 1/5.
Use the logarithm on an interval excluding x=1/5.
Answer:
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
If a retest exposed the same error, rewrite the first incorrect step and explain its correction aloud. If both were independent, return to a mixed exercise or a missed paper question. A parent can ask what changed in the method rather than only asking for the answer.
No. This is a focused substitution set. Some integrals are better handled by parts, partial fractions or standard forms. First inspect the structure rather than forcing one method.
Yes. Use the two print buttons above. The question sheet leaves working space and hides solutions; the worked-solutions option includes the solution steps. Your browser can save either view as a PDF.
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