Find: ((x-3))/((x-1)³)e^x dx

MathsCBSE Class 12 · Sample paper, session 2025-26 · 2 marks · Very short answer

CBSE Class 12 Mathematics sample paper 2025-26: question 23(A)

From the official CBSE sample question paper for 2025-26, section B, question 23(A), for 2 marks. Try it first; the hints and our step-by-step solution are below.

Q23(A) · 2 marks · Very short answerIntegrals

Find: $\int \frac{( x - 3 )}{( x - 1 ) ^{3}} e^{x} d x$

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

Open one hint at a time and try again before the next.

Hint 1

Try writing the numerator $x - 3$ in terms of $x - 1$ .

Hint 2

Check whether one of the two resulting terms is the derivative of the other.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Write $x - 3 = (x - 1) - 2$ : $\int \frac{( x - 3 ) e ^{x}}{( x - 1 ) ^{3}} d x = \int e^{x} [\frac{1}{( x - 1 ) ^{2}} - \frac{2}{( x - 1 ) ^{3}}] d x$ . With $f (x) = \frac{1}{( x - 1 ) ^{2}}$ , $f^{'} (x) = - \frac{2}{( x - 1 ) ^{3}}$ , so the integrand is $e^{x} [f (x) + f^{'} (x)]$ .
1 mark
Since $\int e^{x} [f (x) + f^{'} (x)] d x = e^{x} f (x) + C$ , the integral $= \frac{e ^{x}}{( x - 1 ) ^{2}} + C$ .
1 mark

Answer: $\frac{e ^{x}}{( x - 1 ) ^{2}} + C$

Where the marks are

Splits integrand into $e^{x} [f (x) + f^{'} (x)]$ form with $f (x) = 1/ (x - 1)^{2}$ (1 mark)
Writes $\frac{e ^{x}}{( x - 1 ) ^{2}} + C$ (1 mark)

Revise the chapter

Chapter: Integrals (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Integrals, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26: every question, by section.

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CBSE Class 12 Mathematics sample paper 2025-26: question 23(A)

Hints

Step-by-step solution

Revise the chapter

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