integrate ((x+4)e^x/(x+10)^7)dx and find dy/dx of x^y+y^x=20?

Answer 1

1st question:
y = x+10
dy = dx

int((y-6)e^(y-10)/y^7)dy = (e^-10)int(e^y/y^6)dy - 6(e^-10)int(e^y/y^7)dy

To the side (for integration by parts for the first integral on the right side):
u = 1/y^6, du = -6/y^7dy
dv = e^ydy, v = e^y

= e^(-10)((e^y/y^6) - int(e^y)(-6/y^7)dy) - 6(e^-10)int(e^y)/(y^7)dy +c
= e^-10(e^y/y^6) + 6e^-10int(e^y/y^7)dy - 6e^-10int(e^y/y^7)dy + c
= e^(y-10)/y^6 + c
= e^x/(x+10) + c

Second question:
Differential on both sides

yx^(y-1)dx + x^y/ln(y)dy + xy^(x-1)dy + y^x/ln(x)dx = 0
dx(yx^(y-1) + y^x/ln(x)) + dy(x^y/ln(y) + xy^(x-1)dy) = 0
dy(x^y/ln(y) + xy^(x-1)) = -dx(yx^(y-1) + y^x/ln(x))
dy/dx = -(yx^(y-1) + y^x/ln(x))/(x^y/ln(y) + xy^(x-1))

You really ought to ask these as separate questions though.

Answer 2

First:
let y = e^x/(x+10)^6
dy = [ (x+10)^6 * e^x - e^x * 6(x+10)^5]/(x+10)^12 dx
dy = (x+10)^5)*e^x [ x+10-6]/(x+10)^12 dx
dy = ((x+4)e^x/(x+10)^7)dx
Integrating both sides
Then integrate ((x+4)e^x/(x+10)^7)dx =y = e^x/(x+10)^6
Second:
Let z = x^y, so : ln z = y ln x
Difrentiating w.r.t x
1/z (dz/dx) = y/x + ln x (dy/dx)
dz/dx = y * x ^(y-1) + x^y *ln x (dy/dx) (1)
Let q = y^x, so : ln q = x ln y
Difrentiating w.r.t x
1/q (dq/dx) = x/y (dy/dx) + ln y
dq/dx = x * y ^(x-1)(dy/dx) + y^x *ln y (2)
So: from (1) & (2)
y * x ^(y-1) + x^y *ln x (dy/dx)+x * y ^(x-1)(dy/dx) + y^x *ln y = 0
dy/dx [x^y *ln x +x * y ^(x-1)] = -[y * x ^(y-1) + y^x *ln y ]
dy/dx = -[y * x ^(y-1) + y^x *ln y]/[x^y *ln x +x * y ^(x-1)]

Answer 3

changing: z = x+10
I = integral{(z-6)e^(z-10)/z^7; dz}
I = integral{z^(-6)e^(z-10); dz} - 6 integral{z^(-7)e^(z-10); dz}
I = I1 + I2

by doing:
u = z^(-6) ==> du = -6z^(-7)
dv = e^(z-10)dz ==> v = e^(z-10)

and working the first integral:
I1 = z^(-6)e^(z-10) + 6 integral{z^(-7)e^(z-10); dz}

then:
I = z^(-6)e^(z-10) + 6 integral{z^(-7)e^(z-10); dz} - 6 integral{z^(-7)e^(z-10); dz}
I = z^(-6)e^(z-10)
I = (x+10)^(-6)e^(x)

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I work later the second one

Answer 4

first part:
(e^x)/(x+10)^6 + C

second part:
(x^y)*(y'*ln(x)+y/x)+(y^x)*(lny + xy'/y)=0

Solved for y':
-y*(2*y*x^y + x*lny*y^x)/(x*(y*x^y*lnx+x*y^x)