H2 Maths Integration by Parts: Show the Steps Behind a Shortcut

Understand what a table is summarising

A JC student on SGExams asked whether the DI or tabular version of integration by parts would be accepted in an examination. A forum reply cannot guarantee how a particular script will be marked. What you can control is knowing the underlying identity, following the question’s instructions and writing enough mathematics to make the reasoning clear.

The 2026 H2 Mathematics 9758 syllabus includes integration by parts. A table can help organise repeated differentiation and integration, but it should not hide a sign error or an unsupported step. Ask your teacher to review a sample of your actual written method rather than relying on a blanket assurance about a shortcut.

Choose the two parts deliberately

The identity is ∫u v′ dx = uv − ∫u′v dx. In an original example, integrate x e^x with respect to x. Choose u = x and v′ = e^x, so u′ = 1 and v = e^x. The remaining integral is simpler because differentiating x has removed the polynomial factor.

Substitution gives ∫x e^x dx = x e^x − ∫e^x dx = x e^x − e^x + C. Write the variable of integration and the constant for an indefinite integral. Differentiate the result: the product-rule terms reduce to x e^x, confirming the calculation.

Track signs when repeating the method

For ∫x² e^x dx, take u = x² and v′ = e^x. The first application gives x²e^x − 2∫x e^x dx. Substituting the earlier result yields x²e^x − 2xe^x + 2e^x + C. The alternating signs come from the repeated subtraction in the identity.

This is a case where a tabular layout may reduce clerical work. Still check the derivatives and successive antiderivatives in each column. If the last derivative is not zero, a remaining integral may still be needed. Do not stop a table simply because it has reached a familiar number of rows.

Check the method matches the integrand

Integration by parts is not automatically the best method whenever two expressions are multiplied. If the integrand contains a function and a suitable multiple of its derivative, substitution may simplify it more directly. Try to describe why your chosen split makes the remaining integral easier before committing to several lines of work.

Logarithms can also fit the identity even when no product is visibly written. For ∫ln x dx on x > 0, regard the integrand as ln x multiplied by 1, choose u = ln x and v′ = 1, and obtain x ln x − x + C. A differentiation check confirms the result.

Keep the written solution reviewable

When a question specifies a method or requests exact working, follow that instruction. For a definite integral, evaluate the antiderivative at both bounds and subtract in the correct order. Check any domain restrictions before using an expression involving logarithms.

Practise one example with the full identity and one with your preferred compact layout. Ask whether another reader can reconstruct every step from the shorter version. If not, add the missing mathematical line. Clear working helps expose mistakes during revision without requiring unsupported claims about guaranteed method marks.

Choose a suitable practice question from our H2 Maths papers, checking the level, year and original instructions.

Sources and further reading

Checked on 7 November 2026. The linked discussion is real; worked examples and practice routines are our original guidance. Older discussions are not evidence of current examination rules.

• Does Cambridge accept the DI/tabular method? — 5 June 2024.
• Official curriculum reference