Show the similarity int(x)/((x^2+a^2)^m)dx=(1)/(2(-m+1)(x^2+a^2)^(m-1))+C using the substitution u = x^2 + a^2. Also check that the integration is correct by deriving the answer.

Armorikam 2020-11-06 Answered
Show the similarity x(x2+a2)mdx=12(m+1)(x2+a2)m1+C using the substitution u=x2+a2. Also check that the integration is correct by deriving the answer.
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Expert Answer

escumantsu
Answered 2020-11-07 Author has 98 answers
Step 1
put x2+a2=u
2xdx=du
xdx=du2
x(x2+a2)mdx=(du2um))
=12umdu
=12um+1m+1+C[xndx=xn+1n+1]
=12(x2+a2)m+1m+1+C
=12(m+1)(x2+a2)m1+C
Veification:
Differentiating obtained integral value with respect to x, we get
ddx(12(m+1)(x2+a2)m1+C)=ddx((x2+a2)m+12(m+1)+C)
=12(m+1)(m+1)(x2+a2)m+11d(x2+a2)dx+0
12(x2+a2)m(2x+0)
=x(x2+a2)m
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