\[Evaluate:\,\,\int {\sqrt {\frac{{1 + \sin 8x}}{{1 - \sin 8x}}} } \,\,dx\]
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; 03-07-2019 03:00 AM
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\[\begin{array}{l}
\left( 1 \right)\,\,\,\,\int_0^a {f\left( x \right)} \,dx = \,\int_0^a {f\left( {a - x} \right)} \,dx\\
\left( 2 \right)\,\,\,\,\int_a^b {f\left( x \right)} \,dx = \,\int_a^b {f\left( {a + b - x} \right)} \,dx\\
\left( 3 \right)\,\,\,\,\int_{ - a}^a {f\left( {{x^2}} \right)} \,dx = \,2\int_0^a {f\left( {{x^2}} \right)} \,dx\\
\left( 4 \right)\,\,\,\,\int_0^{\pi /2} {{{\sin }^2}x} \,dx = \,\,\int_0^{\pi /2} {{{\cos }^2}x} \,dx
\end{array}\]
\[\begin{array}{l}
\left( 2 \right)\,\,\,\int\limits_a^b {f\left( {a + b - x} \right)} \,\,dx = \int\limits_a^b {f\left( x \right)} \,\,dx\\
The\,\,\Pr ove\,:\\
Putting\,\,a + b - x = y\,\,\,then\,\,\, - dx = dy\,\,\,,\,\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,wh en\,\,x = a \Rightarrow y = b\,\,and\,\,\,when\,\,x = b \Rightarrow y = a\\
\,\,\,\int\limits_a^b {f\left( {a + b - x} \right)} \,\,dx = \,\int\limits_b^a {f\left( y \right)} \,\left( {\, - dy} \right) = \underbrace {\int\limits_a^b {f\left( y \right)} \,\,dy = \int\limits_a^b {f\left( x \right)} \,\,dx}_{Changing\,y\,to\,x}
\end{array}\]
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