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  • Evaluate the definite integrals in Exercises 1 to 20. Integral 0 to pi sin^2 x/2 - cos^2 x/\2 dx

Evaluate the definite integrals in Exercises 1 to 20.

    Q18.    \int^\pi_0(\sin^2\frac{x}{2} - \cos^2\frac{x}{2})dx

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Given integral: \int^\pi_0(\sin^2\frac{x}{2} - \cos^2\frac{x}{2})dx

Consider the integral \int (\sin^2\frac{x}{2} - \cos^2\frac{x}{2})dx

can be rewritten as: -\int (\cos^2\frac{x}{2} - \sin^2\frac{x}{2})dx = -\int_0^{\pi} \cos x dx

                                                                                    = \sin x

So, we have the function of x, f(x) =\sin x

Now, by Second fundamental theorem of calculus, we have

I = f(\pi) - f(0)

\Rightarrow \sin \pi - \sin 0 = 0-0 =0

Posted by

Divya Prakash Singh

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