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Integrals

Short Answer Type

251.

Find space integral fraction numerator dx over denominator straight x squared space plus 4 straight x space plus 8 end fraction

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252.
Find $integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis squared end fraction dx$

Let space straight x squared space equals space straight y rightwards double arrow space 2 xdx space equals space dy rightwards double arrow space dx space equals space fraction numerator dy over denominator 2 straight x end fraction integral fraction numerator dy over denominator left parenthesis straight y plus 1 right parenthesis left parenthesis straight y plus 2 right parenthesis squared end fraction Let space fraction numerator 1 over denominator left parenthesis straight y plus 1 right parenthesis left parenthesis straight y plus 2 right parenthesis squared end fraction space equals space fraction numerator straight X over denominator straight y plus 1 end fraction space plus fraction numerator straight Y over denominator straight y plus 2 end fraction space plus fraction numerator straight Z over denominator left parenthesis straight y plus 2 right parenthesis squared end fraction...... space left parenthesis 1 right parenthesis

⇒ 1 = X (y+2)2 + Y(y+1)(y+2)+C(y+1).....(2)
Putting y = -2 in (2)
1 = Z(-2+1)
⇒Z = -1
Putting y = -1 in (2)
1 = X(-1 +22)
⇒1 = A (1)
⇒A =1
Putting y =0 in (2)
1 = 4X + Y(2) +Z
⇒ 1=4 + 2Y-1
⇒ 1 =3 +2Y
⇒ -2 = 2Y
⇒ Y = -1
$fraction numerator 1 over denominator left parenthesis straight y plus 1 right parenthesis left parenthesis straight y plus 2 right parenthesis squared end fraction space equals space fraction numerator 1 over denominator straight y plus 1 end fraction minus fraction numerator 1 over denominator straight y plus 2 end fraction minus fraction numerator 1 over denominator left parenthesis straight y plus 2 right parenthesis squared end fraction rightwards double arrow space integral fraction numerator dy over denominator left parenthesis straight y plus 1 right parenthesis left parenthesis straight y plus 2 right parenthesis squared end fraction space equals space integral fraction numerator begin display style dy end style over denominator straight y plus 1 end fraction minus integral fraction numerator dy over denominator straight y plus 2 end fraction minus integral fraction numerator dy over denominator left parenthesis straight y plus 2 right parenthesis squared end fraction equals log space vertical line straight y plus 1 vertical line minus log space vertical line straight y plus 2 vertical line space plus fraction numerator 1 over denominator straight y plus 2 end fraction space plus straight C Hence comma space integral fraction numerator 2 straight x over denominator left parenthesis straight x squared plus 1 right parenthesis left parenthesis straight x squared plus 2 right parenthesis squared end fraction dx space equals space log space vertical line straight x squared plus 2 vertical line space plus fraction numerator 1 over denominator straight x squared plus 2 end fraction plus straight C$

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253.

Evaluate: $integral subscript 0 superscript straight pi fraction numerator straight x space sin space straight x over denominator 1 plus cos squared x end fraction dx$