Integration by parts: Class 12 practice

What you should know first

• Standard integrals
• Product rule

The idea behind integration by parts

∫u dv = uv − ∫v du. Choose u to be the part that becomes simpler when differentiated (often x, or a logarithm), using the LIATE order as a guide.

Step-by-step method

1. Choose u and dv.
2. Find du and v.
3. Apply uv − ∫v du.
4. Simplify; add + C.

Worked example 1 (Basic)

Find ∫ x e^(4x) dx.

1. u = x, dv = e^(4x) dx → v = e^(4x)/4
2. = x e^(4x)/4 − ∫ e^(4x)/4 dx
3. Answer: e^(4x)(x/4 − 1/16) + C

Worked example 2 (Standard)

Find ∫ x cos(2x) dx.

1. u = x, v = sin(2x)/2
2. = x sin(2x)/2 − ∫ sin(2x)/2 dx
3. Answer: x sin(2x)/2 + cos(2x)/4 + C

Worked example 3 (Challenge)

Find ∫ x^2 ln x dx.

1. u = ln x, dv = x^2 dx → v = x^3/3
2. = x^3 ln x/3 − ∫ x^2/3 dx
3. Answer: x^3(ln x/3 − 1/9) + C

Common mistakes and misconceptions

• Choosing u = eˣ and making it harder
• Sign error in − ∫v du

Practice questions

1. (Basic) Find ∫ x e^(2x) dx.
2. (Basic) Find ∫ x e^(1x) dx.
3. (Standard) Find ∫ x cos(4x) dx.
4. (Standard) Find ∫ x cos(1x) dx.
5. (Challenge) Find ∫ x^1 ln x dx.
6. (Challenge) Find ∫ x^4 ln x dx.

Answer key with full working

• Q1: u = x, dv = e^(2x) dx → v = e^(2x)/2 → = x e^(2x)/2 − ∫ e^(2x)/2 dx → Answer: e^(2x)(x/2 − 1/4) + C
• Q2: u = x, dv = e^(1x) dx → v = e^(1x)/1 → = x e^(1x)/1 − ∫ e^(1x)/1 dx → Answer: e^(1x)(x/1 − 1/1) + C
• Q3: u = x, v = sin(4x)/4 → = x sin(4x)/4 − ∫ sin(4x)/4 dx → Answer: x sin(4x)/4 + cos(4x)/16 + C
• Q4: u = x, v = sin(1x)/1 → = x sin(1x)/1 − ∫ sin(1x)/1 dx → Answer: x sin(1x)/1 + cos(1x)/1 + C
• Q5: u = ln x, dv = x^1 dx → v = x^2/2 → = x^2 ln x/2 − ∫ x^1/2 dx → Answer: x^2(ln x/2 − 1/4) + C
• Q6: u = ln x, dv = x^4 dx → v = x^5/5 → = x^5 ln x/5 − ∫ x^4/5 dx → Answer: x^5(ln x/5 − 1/25) + C

Syllabus note

This skill is listed under "Integrals" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Integrals chapter notes (Class 12)
• Definite integrals of polynomials
• Integration by substitution
• Properties of definite integrals
• Derivatives of polynomials and slope of a tangent
• Evaluating 2×2 and 3×3 determinants
• Matrix multiplication
• Conditional probability and Bayes' theorem
• All Class 12 practice

Frequently asked questions

Which chapter is integration by parts in?

"Integrals", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.