Question

Evaluate
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx plus square root of 3 cosx end fraction dx$

Solution

Let I = $integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx plus square root of 3 cosx end fraction dx$
Put    $1 space equals straight r space cos space straight alpha$ ...(1)
and    $square root of 3 space equals space straight r space sin space straight alpha$ ...(2)
Squaring and adding (1) and (2), we get
$4 space equals space straight r squared$ $rightwards double arrow space space space straight r space equals space 2$
Dividing (2) by (1), $tan space straight alpha space equals space square root of 3$ $rightwards double arrow space space space straight a space equals space straight pi over 3$
$therefore$ $straight I space equals space 1 over straight r integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx space cosα space plus space cosx space sinα end fraction dx space equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sin left parenthesis straight x plus straight alpha right parenthesis end fraction dx$
   $equals space 1 half integral subscript 0 superscript straight pi over 2 end superscript cosec left parenthesis straight x plus straight alpha right parenthesis space dx space equals space 1 half open square brackets negative log space open vertical bar cosec left parenthesis straight x plus straight alpha right parenthesis plus cot left parenthesis straight x plus straight a right parenthesis close vertical bar close square brackets subscript 0 superscript straight pi over 2 end superscript$
$equals space minus 1 half open square brackets log space open vertical bar cosec space open parentheses straight pi over 2 plus straight pi over 3 close parentheses plus cot space open parentheses straight pi over 2 plus straight pi over 3 close parentheses close vertical bar minus log open vertical bar cosec straight pi over 3 plus cot straight pi over 3 close vertical bar close square brackets$
$equals negative 1 half open square brackets log space open vertical bar sec straight pi over 3 minus tan straight pi over 3 close vertical bar minus log open vertical bar cosec straight pi over 3 plus cot straight pi over 3 close vertical bar close square brackets$
   $equals negative 1 half open square brackets log space open vertical bar 2 minus square root of 3 close vertical bar minus log space open vertical bar fraction numerator 2 over denominator square root of 3 end fraction plus fraction numerator 1 over denominator square root of 3 end fraction close vertical bar close square brackets equals negative 1 half open square brackets log left parenthesis 2 minus square root of 3 right parenthesis space minus space log square root of 3 close square brackets equals space minus 1 half log open parentheses fraction numerator 2 minus square root of 3 over denominator square root of 3 end fraction close parentheses$

For Daily Free Study Material Join wiredfaculty

Integrals

Evaluate
$integral subscript 0 superscript straight pi over 2 end superscript fraction numerator 1 over denominator sinx plus square root of 3 cosx end fraction dx$

Some More Questions From Integrals Chapter

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Phone Number

Email Address

Loaction