3.76 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS
Calculus
Example 158:elog x dx = logx.1.dx
logx dx dlogx dx dx
dx
= −^
(^)
logx.x^1 .x dx xlogx dx xlogx x c.
x
= −^ = − = − +
Example 159:Evaluate : e logx^2 dx
I logx .1.dx logx. 1.dx^22 dlogx. 1dx dx^2
dx
= = −^
(^)
logx .x^222 x.x dx xlogx 2 dx xlogx 2x c^22
x
= −^ = − = − +
Example 160:Evaluate : e ex (1 + x) log (xex)dx.
Let xex = u, (ex + xex)dx = du or, ex (1 + x) dx = du.
Integral (i) = elog u du = u log u – u =xe log(xe ) xe cx x− x+
Example 161:Evaluate : e(x^2 – 2x + 5)e–x dx.
I = e x^2 e–xdx – 2 e xe–x dx + 5ee–x dx
x e dx^2 x dx e dx dx 2 xe dx 5e ( 1)^2 x x x
dx
= − −^ − − − + − −
=x .e ( 1) 2 xe .( 1)dx 2 xe dx 5e2 x− − − x − − −x − −x
= −x e 2 xe dx 2 xe dx 5e x e 5e c2 x− + −x − −x − −x= − 2 x− − −x+
Standard Integrals :
- ( )
2 2 x x a a^22222
x a dx+ = 2 + + 2 log x x a+ +
2. ( ) ( )
(^222)
x a dx^22 x x a a log x x a.^22
2 2
−
− = − + −
Example 162:Find the value of 25x 16 dx^2 +
I 5x 4 dx.= ( )^2 +^2 Let 5x = u, 5dx = du