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Integrals

Short Answer Type

41.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space sin space 2 straight x space dx$

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42.

Evaluate the following definite integral
$integral subscript straight pi over 6 end subscript superscript straight pi over 4 end superscript space cosec space straight x space dx$

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43.

Evaluate the following definite integral
$integral subscript 4 superscript 5 straight e to the power of straight x dx$

128 Views

44.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript space left parenthesis 2 sec squared straight x plus straight x cubed plus 2 right parenthesis dx$

93 Views

45.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi over 4 end superscript left parenthesis 2 sec squared straight x plus straight x cubed plus 2 right parenthesis space dx$

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46.

Evaluate the following definite integral
$integral subscript 0 superscript straight pi open parentheses sin squared straight x over 2 minus cos squared straight x over 2 close parentheses space dx$

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47.

Evaluate the following definite integral
$integral subscript 1 superscript 2 left parenthesis 4 straight x cubed minus 5 straight x squared plus 6 straight x plus 9 right parenthesis space dx.$

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48.

Evaluate $integral subscript 0 superscript 1 fraction numerator dx over denominator square root of 1 plus straight x end root minus square root of straight x end fraction.$

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49.
Evaluate $integral subscript 0 superscript 1 space straight x space straight e to the power of straight x squared space dx$

Let I = $integral subscript 0 superscript 1 space straight x space straight e to the power of straight x squared end exponent space dx$
Let $straight I subscript 1 space equals space integral space straight x space straight e to the power of straight x squared end exponent space dx$
Put $straight x squared space equals space straight y comma space space$ $therefore space space space 2 space straight x space dx space equals space dy$ $rightwards double arrow$ $straight x space dx space equals space 1 half dx$
$therefore$ $straight I subscript 1 space equals space 1 half integral straight e to the power of straight y dy space equals space 1 half straight e to the power of straight y space equals space 1 half straight e to the power of straight x squared end exponent space equals space straight F left parenthesis straight x right parenthesis comma space say$
$therefore$ by second fundamental theorem,
I = F(1) - F(0) = $1 half straight e to the power of 1 minus 1 half straight e to the power of 0 space equals space 1 half straight e minus 1 half space equals space 1 half left parenthesis straight e minus 1 right parenthesis$