Evaluate : 
 
1. integral subscript 0 superscript straight pi divided by 4 end superscript log left parenthesis 1 plus tan theta right parenthesis d theta
2. integral subscript 1 superscript 2 c os open parentheses πx divided by 2 close parentheses d x

Asked by Navya Benny | 24th Dec, 2015, 04:00: PM

Expert Answer:

1. space I equals integral subscript 0 superscript pi over 4 end superscript log open parentheses 1 plus tan theta close parentheses d theta equals integral subscript 0 superscript pi over 4 end superscript l o g open parentheses 1 plus t a n open parentheses pi over 4 minus theta close parentheses close parentheses d theta equals integral subscript 0 superscript pi over 4 end superscript l o g open parentheses 1 plus fraction numerator 1 minus tan theta over denominator 1 plus tan theta end fraction close parentheses d theta equals integral subscript 0 superscript pi over 4 end superscript l o g open parentheses fraction numerator 2 over denominator 1 plus t a n theta end fraction close parentheses d theta  rightwards double arrow 2 I equals integral subscript 0 superscript pi over 4 end superscript l o g open parentheses 1 plus t a n theta close parentheses d theta space plus space integral subscript 0 superscript pi over 4 end superscript l o g open parentheses fraction numerator 2 over denominator 1 plus t a n theta end fraction close parentheses d theta rightwards double arrow 2 I equals integral subscript 0 superscript pi over 4 end superscript open square brackets log open parentheses 1 plus tan theta close parentheses plus l o g open parentheses fraction numerator 2 over denominator 1 plus t a n theta end fraction close parentheses close square brackets d theta rightwards double arrow 2 I equals integral subscript 0 superscript pi over 4 end superscript open square brackets l o g open parentheses 1 plus t a n theta close parentheses open parentheses fraction numerator 2 over denominator 1 plus t a n theta end fraction close parentheses close square brackets d theta rightwards double arrow 2 I equals integral subscript 0 superscript pi over 4 end superscript log 2 space d theta rightwards double arrow 2 I equals log 2 cross times pi over 4 rightwards double arrow I equals pi over 8 log 2
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Answered by satyajit samal | 27th Dec, 2015, 06:26: AM

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