medium difficulty7 min

Integration by Parts

Integration by parts uses the formula ∫u dv=uv-∫v du to integrate a product of two functions, with u chosen so that its derivative becomes simpler.

In one line: First function as u, second function integrated as v, then subtract the new integral.

Practice This Concept

Understand it

This method is useful when the integrand is a product such as x e^x, x sinx, logx, or inverse trigonometric functions. The choice of u matters; the ILATE order often helps choose u: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.

Example

For ∫x e^x dx, take u=x and dv=e^x dx. Then du=dx and v=e^x, so the integral is xe^x-∫e^x dx=xe^x-e^x+C.

Formula and rule

Formula: ∫u dv=uv-∫v du.

Conditions: the integrand should be expressible as a product u·dv; v must be integrable; du should simplify the next integral.

For ∫logx dx or ∫tan⁻¹x dx, take dv=dx.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Evaluate ∫x cosx dx.

1Let u=x, dv=cosx dx.
2Then du=dx and v=sinx.
3Using ∫u dv=uv-∫v du, integral =x sinx-∫sinx dx=x sinx+cosx+C.
4Check: derivative of x sinx+cosx is sinx+xcosx-sinx=xcosx.

Method you can reuse

1Identify the product.
2Choose u using ILATE and ease of differentiation.
3Put the remaining factor with dx as dv.
4Find du and v.
5Substitute into uv-∫vdu.
6Evaluate the remaining integral.
7Differentiate the final answer to check signs.

Visual tip

No diagram is required. A two-column layout for u, dv, du, and v is the most helpful written support.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In ∫x sinx dx, what should be chosen as u by ILATE?

Think first, then show answer

Choose u=x, because algebraic functions come before trigonometric functions in ILATE.

Common wrong answer

Writing ∫u dv=uv+∫vdu instead of uv-∫vdu, causing a sign error in nearly every by-parts solution.

Don't mix this up

A common mistake is choosing u poorly, making the remaining integral harder instead of simpler.

Exam tip

Use ILATE as a guide, but still check whether differentiating u simplifies the expression and integrating dv is easy.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Common in long-answer questions involving products, logarithmic functions, inverse trigonometric functions, and repeated application such as ∫e^x sinx dx.

1
Enough for 1 mark
Integration by parts uses the formula ∫u dv=uv-∫v du to integrate a product of two functions, with u chosen so that its derivative becomes simpler.
2
Add for 2 marks
Example: For ∫x e^x dx, take u=x and dv=e^x dx. Then du=dx and v=e^x, so the integral is xe^x-∫e^x dx=xe^x-e^x+C.
3
Add for 3 marks
This method is useful when the integrand is a product such as x e^x, x sinx, logx, or inverse trigonometric functions.

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