Paper 4: Fundamentals of Business Mathematics & Statistic

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 :

1. ( )
   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