medium difficulty6 min

Standard Integrals of Particular Forms

Integrals of particular forms are standard results for expressions such as 1/(x²+a²), 1/(x²-a²), 1/√(a²-x²), and related algebraic forms.

In one line: Plus with squares often leads to tan inverse; square root of a² minus x² leads to sine inverse.

Practice This Concept

Understand it

These forms are solved by recognizing the exact structure and matching it with the correct standard formula. The sign between terms and the square-root placement decide whether the result involves tan inverse, sine inverse, logarithm, or another standard expression.

Example

∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C for a>0.

Formula and rule

Key forms for a>0: ∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C; ∫dx/√(a²-x²)=sin⁻¹(x/a)+C; ∫dx/(x²-a²)=(1/(2a))ln|(x-a)/(x+a)|+C; ∫dx/(a²-x²)=(1/(2a))ln|(a+x)/(a-x)|+C.

Conditions and signs must be checked.

See it worked out

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

Worked example

Evaluate ∫dx/(x²+25).

1Here x²+25=x²+5².
2Using ∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C, the answer is (1/5)tan⁻¹(x/5)+C.

Method you can reuse

1Identify the exact form.
2Rewrite constants as squares.
3Check whether the expression is x²+a², a²-x², x²-a², or a²+x².
4Apply the matching formula.
5Include absolute value in logarithmic forms where required.
6Add +C for indefinite integrals.

Visual tip

A diagram is generally not needed. Formula recognition, sign checking, and constant-square rewriting are the useful supports.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Evaluate ∫dx/√(16-x²).

Think first, then show answer

sin⁻¹(x/4)+C, using ∫dx/√(a²-x²)=sin⁻¹(x/a)+C with a=4.

Common wrong answer

For ∫dx/(x²-9), writing (1/3)tan⁻¹(x/3)+C instead of the logarithmic form because the minus sign was ignored.

Don't mix this up

A frequent mistake is confusing x²+a² with a²-x² and using tan inverse when sine inverse is required.

Exam tip

Match the denominator exactly before applying a formula. Rewrite constants as squares whenever possible, such as x²+9=x²+3².

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

Often asked as direct standard-form questions or after completing the square. May also appear inside substitution-based integrals where the transformed integral matches a standard form.

1
Enough for 1 mark
Integrals of particular forms are standard results for expressions such as 1/(x²+a²), 1/(x²-a²), 1/√(a²-x²), and related algebraic forms.
2
Add for 2 marks
These forms are solved by recognizing the exact structure and matching it with the correct standard formula.
3
Add for 3 marks
The sign between terms and the square-root placement decide whether the result involves tan inverse, sine inverse, logarithm, or another standard expression.

MCQ Quiz

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