Math, asked by PragyaTbia, 1 year ago

Evaluate the definite integrals: \int ^\pi_0 {{(sin^2 \frac{x}{2} - cos^2 \frac {x}{2}}) \, dx

Answers

Answered by MaheswariS
0

Answer:

0

Step-by-step explanation:

Concept:

I have applied decomposition method to solve this problem.

In decomposition method, the given non-integrable function is decomposed into integrable function by using algebraic identities, trigonometric identities, etc.

Formula used:

cosA=cos^2\frac{A}{2}-sin^2\frac{A}{2}

Now,

\int\limits^{\pi}_0[sin^2{\frac{x}{2}}-cos^2{\frac{x}{2}}]\;dx\\\\=-\int\limits^{\pi}_0[cos^2{\frac{x}{2}}-sin^2{\frac{x}{2}}]\;dx\\\\=-\int\limits^{\pi}_0[cosx]\;dx\\\\=-[sinx]^{\pi}_0\\\\=-[sin{\pi}-sin0]\\\\=-[0-0]\\\\=0

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