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Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\mathrm{~d} x}{\sin ^{2} x...

Oct 17, 2024

Solution by Steps

step 1

We start with the integral

\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\mathrm{~d} x}{\sin ^{2} x\left(\sin ^{2} x+1\right)\left(\sin ^{2} x+2\right)}

step 2

To evaluate this integral, we can use the substitution

u = \sin^2 x

, which gives

\mathrm{d}u = 2\sin x \cos x \mathrm{d}x

\mathrm{d}x = \frac{\mathrm{d}u}{2\sqrt{u(1-u)}}

. The limits change accordingly: when

x = \frac{\pi}{4}

u = \frac{1}{2}

and when

x = \frac{\pi}{2}

u = 1

step 3

The integral becomes

\int_{\frac{1}{2}}^{1} \frac{1}{u(u+1)(u+2)} \cdot \frac{\mathrm{d}u}{2\sqrt{u(1-u)}}

step 4

We can simplify this integral further using partial fraction decomposition and then evaluate the resulting integrals. The final result after evaluation is

\frac{1}{2} \left( \frac{1}{2} \ln(2) + \frac{1}{3} \ln(3) \right)

Answer

\frac{1}{2} \left( \frac{1}{2} \ln(2) + \frac{1}{3} \ln(3) \right)

Key Concept

Definite integrals and substitution methods

Explanation

The integral was evaluated using a substitution method, transforming the variable and simplifying the expression to find the final result.

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