The following anti derivatives. Please show work, be careful with the coefficien

Question

The following anti derivatives. Please show work, be careful with the coefficient the question and the answer, and put no more than two answers per page. u subs: integral log(x)/x dx integral x dx/Squareroot 1 - x^4 integral e^Squareroot x/Squareroot x dx x Squareroot 1 - x^2 dx parts: integral x^2 sin(x) dx integral x^3 log(x) dx integral e^x sin(x)dx integral log(log(x))/x dx trig sub: integral dx/Squareroot x^2 - 1 integral Squareroot 1 - x^2 dx Partial fraction etc: integral dx/(x + a)(x + b) integral x^6/x - 1 dx integral 1 + x/x - x^2 dx integral 5x - 13/x^2 - 4x - 5 dx on you (mostly y-subs of your choice)

Explanation / Answer

1)

(log(x))/x dx

substitute log(x)=u

differentiate =>(1/x)dx =du

(log(x))/x dx

=u du

=(1/2)u2 +C

=(1/2)(log(x))2 +C

================================================

2) xdx/(1-x4) = xdx/(1-(x2)2)

substitute x2=u

differentiate => 2x dx =du

=> x dx =(1/2) du

xdx/(1-x4) =(1/2) du/(1-u2)

=> (1/2)sin-1(u) +C

=> (1/2)sin-1(x2) +C

xdx/(1-x4) =(1/2)sin-1(x2) +C

===================================

3) ex/x dx

substitute x =u

differentiate =>(1/2x)dx =du

=>(1/x)dx =2du

ex/x dx

= eu *2du

=2eu +C

=2ex +C

ex/x dx =2ex +C

===========================================

4) x(1-x2) dx

substitute 1-x2=u

differentiate =>0-2x dx =du

=>x dx =-(1/2)du

x(1-x2) dx

= u (-(1/2)du)

= -(1/2)(2/3)u3/2 +C

= -(1/3)u3/2 +C

= -(1/3)(1-x2)3/2 +C

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