Evaluate the definite integral from 4 to 0.?

Question

Evaluate the definite integral from 4 to 0.
{dx} / {2 x + 7}

Answer 1

hint: it involves a natural logarithm

Answer 2

Remember that the derivative of lnx is 1/x so the integral of 1/x is lnx.
So
∫(4 to 0) 1/(dx+7) dx = ln(2x+7) / 2 (from 4 to 0)
Note the divide by two as if you went backwards and took the derivative you would have to multiple by two from infront of the x due to the chain rule.
Then subbing in the limits
∫(4 to 0) 1/(dx+7) dx = ln(8+7) / 2 - ln(7)/2
∫(4 to 0) 1/(dx+7) dx = ln(15) / 2 - ln(7)/2
∫(4 to 0) 1/(dx+7) dx = ln(15 / 7)/2

Answer 3

antiderivative:

.5ln(2x + 7) + C

Evaluating this from 0 to 4 (when you say "evaluated from" you say the limit on the lower end of the integral before the one on top) gives you .5(ln15-ln7) which equals
.5(ln(15/7)).

Answer 4

∫dx/(2x+7)

=>1/2∫d(2x+7)/(2x+7)

=>(1/2) ln(2x + 7)

applying limits 4 to 0

(1/2)[ln(0+7) - ln(8+7)]

(1/2)[ln(7) - ln(15)]

(1/2)[-0.762] = -(0.381)