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Integration Using Trigonometric Identities

Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas.

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Student-friendly explanation

Direct integration may not be possible when powers or products of trigonometric functions appear. Identities convert the integrand into sums or standard forms. The chosen identity must match the power, product, or angle structure in the question.

How to write this in exams

  1. 1

    Start with the exact idea

    Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas.

  2. 2

    Then show how to use it

    Inspect whether the integrand is a square, product, or expression involving tan and sec. Choose a matching identity. Rewrite the full integrand. Integrate term-wise. Check angle multipliers such as 2x or ax. Add +C when indefinite.

  3. 3

    Add one concrete example

    ∫sin^2x dx=∫(1-cos2x)/2 dx=x/2-sin2x/4+C.

  4. 4

    Avoid this incomplete answer

    Writing ∫cos2x dx=sin2x+C instead of sin2x/2+C, missing the factor from the derivative of 2x.

Definition

Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas.

Example

∫sin^2x dx=∫(1-cos2x)/2 dx=x/2-sin2x/4+C.

Rule to remember

Useful identities: sin^2x=(1-cos2x)/2, cos^2x=(1+cos2x)/2, 2sinxcosx=sin2x, sin^2x+cos^2x=1, 1+tan^2x=sec^2x. Apply identities before integration when the original form is not standard.

Memory hook

First reduce the trig expression, then integrate the reduced form.

Examples and method

Worked example

Evaluate ∫sinx cosx dx. Use 2sinxcosx=sin2x, so sinxcosx=(1/2)sin2x. Integral =(1/2)∫sin2x dx=(1/2)(-cos2x/2)+C=-cos2x/4+C. Equivalent answer: sin^2x/2+C.

Method to apply

Inspect whether the integrand is a square, product, or expression involving tan and sec. Choose a matching identity. Rewrite the full integrand. Integrate term-wise. Check angle multipliers such as 2x or ax. Add +C when indefinite.

Diagram support

No diagram is needed. The essential support is a list of identities and careful angle handling, especially when 2x appears.

How CBSE asks it

Questions may ask direct evaluation, simplification before integration, or comparison of equivalent answers. Long-answer questions may combine identities with substitution.

Avoid common mistakes

Common confusion

Students sometimes integrate sin^2x as if it were sin x, which is incorrect because powers of trigonometric functions need reduction or another suitable identity.

Common wrong answer

Writing ∫cos2x dx=sin2x+C instead of sin2x/2+C, missing the factor from the derivative of 2x.

Exam tip

Before integrating a trigonometric power or product, ask whether an identity can reduce it to first powers or a standard derivative pair.

Quick check

Find ∫cos^2x dx.

Using cos^2x=(1+cos2x)/2, integral =x/2+sin2x/4+C.

Answer writing and exam use

1-mark answer

Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas.

2-mark answer

Some trigonometric integrals are first simplified using identities, such as sin^2x=(1-cos2x)/2 and cos^2x=(1+cos2x)/2, before applying standard integration formulas. Useful identities: sin^2x=(1-cos2x)/2, cos^2x=(1+cos2x)/2, 2sinxcosx=sin2x, sin^2x+cos^2x=1, 1+tan^2x=sec^2x. Apply identities before integration when the original form is not standard. ∫sin^2x dx=∫(1-cos2x)/2 dx=x/2-sin2x/4+C.

3-mark answer

Direct integration may not be possible when powers or products of trigonometric functions appear. Identities convert the integrand into sums or standard forms. The chosen identity must match the power, product, or angle structure in the question. Useful identities: sin^2x=(1-cos2x)/2, cos^2x=(1+cos2x)/2, 2sinxcosx=sin2x, sin^2x+cos^2x=1, 1+tan^2x=sec^2x. Apply identities before integration when the original form is not standard. Evaluate ∫sinx cosx dx. Use 2sinxcosx=sin2x, so sinxcosx=(1/2)sin2x. Integral =(1/2)∫sin2x dx=(1/2)(-cos2x/2)+C=-cos2x/4+C. Equivalent answer: sin^2x/2+C. Questions may ask direct evaluation, simplification before integration, or comparison of equivalent answers. Long-answer questions may combine identities with substitution. Writing ∫cos2x dx=sin2x+C instead of sin2x/2+C, missing the factor from the derivative of 2x.
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