Evaluate the following1.int e^x sqrt (1-e^(2x)) dx2. int frac{x^2}{sqrt 5+x^2}dx

Harlen Pritchard 2021-08-18 Answered
Evaluate the following
1.ex1e2xdx
2. x25+x2dx
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Expert Answer

pattererX
Answered 2021-08-19 Author has 95 answers
1.Finding the integral ex1e2xdx.
Let us first apply substitution u=ex we get
ex1e2xdx=1u2du
Further apply the substitution u=sinv we get
1u2du=cos2(v)dv
=1+cos(2v)2dv[:cos2(x)=1+cos(2x)2]
= 12(v+12sin(2v))
Substitute back v=arcsin(u), and u=ex, we get v=arcsin(ex),we get ex1e2xdx=12(arcsin(ex)+12sin(2arcsin(ex)))+C
2. Finding the integral x25+x2dx
x25+x2dx=5tan2(u)sec(u)du[x=5tan(u)]
=5(1+sec2(u))sec(u)du[tan2(x)=1+sec2(x)]
=5(ln|tan(u)+sec(u)|+12sec(u)tan(u)+12lntan(u)+sec)
=

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