Integration by substitution: Class 12 practice
If the integrand contains a function and (a multiple of) its derivative, substitute u for the inner function. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Standard integrals
- Chain rule
The idea behind integration by substitution
If the integrand contains a function and (a multiple of) its derivative, substitute u for the inner function. Then ∫f(g(x))g′(x) dx = ∫f(u) du.
Step-by-step method
- Choose u = inner function.
- Find du in terms of dx.
- Rewrite and integrate in u.
- Substitute back; add + C.
Worked example 1 (Basic)
Find ∫ 2x(1x² + 9)^4 dx.
- u = 1x² + 9, du = 2x dx
- ∫u^4 du = u^5/5
- Answer: (1x² + 9)^5/5 + C
Worked example 2 (Standard)
Find ∫ x/(x² + 9) dx.
- u = x² + 9, du = 2x dx
- = ½ ln|u|
- Answer: ½ ln(x² + 9) + C
Worked example 3 (Challenge)
Find ∫ cos(1x) e^(sin 1x) dx.
- u = sin 1x, du = 1 cos 1x dx
- = (1/1) e^u
- Answer: (1/1) e^(sin 1x) + C
Common mistakes and misconceptions
- Not adjusting for the constant in du
- Forgetting to substitute back
Practice questions
- (Basic) Find ∫ 8x(4x² + 7)^5 dx.
- (Basic) Find ∫ 4x(2x² + 2)^5 dx.
- (Standard) Find ∫ x/(x² + 1) dx.
- (Standard) Find ∫ x/(x² + 6) dx.
- (Challenge) Find ∫ cos(4x) e^(sin 4x) dx.
- (Challenge) Find ∫ cos(3x) e^(sin 3x) dx.
Answer key with full working
- Q1: u = 4x² + 7, du = 8x dx → ∫u^5 du = u^6/6 → Answer: (4x² + 7)^6/6 + C
- Q2: u = 2x² + 2, du = 4x dx → ∫u^5 du = u^6/6 → Answer: (2x² + 2)^6/6 + C
- Q3: u = x² + 1, du = 2x dx → = ½ ln|u| → Answer: ½ ln(x² + 1) + C
- Q4: u = x² + 6, du = 2x dx → = ½ ln|u| → Answer: ½ ln(x² + 6) + C
- Q5: u = sin 4x, du = 4 cos 4x dx → = (1/4) e^u → Answer: (1/4) e^(sin 4x) + C
- Q6: u = sin 3x, du = 3 cos 3x dx → = (1/3) e^u → Answer: (1/3) e^(sin 3x) + 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
Frequently asked questions
Which chapter is integration by substitution 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.