Integration and Its Applications
What the exam tests
Integration questions test antiderivatives, definite integrals, area under a curve, and simple economic applications such as total cost from marginal cost. The exam may also ask for a constant of integration or use substitution.
Core rules
- (\int x^n,dx=\frac{x^{n+1}}{n+1}+C) for (n\ne-1).
- (\int \frac1x,dx=\ln|x|+C), (\int e^x,dx=e^x+C), (\int \sin x,dx=-\cos x+C), (\int \cos x,dx=\sin x+C).
- (\int_a^b f(x),dx=F(b)-F(a)), where (F'(x)=f(x)).
- For substitution, change the variable and, for a definite integral, change the limits.
- Area between a curve and the x-axis is found by integrating (|f(x)|); split the interval where the curve crosses the axis.
Worked example
Evaluate (\int_1^3(2x+1),dx). An antiderivative is (x^2+x). Applying the limits: ((3^2+3)-(1^2+1)=(9+3)-(1+1)=12-2=10). The indefinite integral would include (+C), but the definite integral is a number.
Traps
Do not forget (+C) in an indefinite integral. The power rule fails at (n=-1). Reverse the limits carefully: (\int_a^b=-\int_b^a). When substituting, do not mix old limits with the new variable. Area is non-negative, so handle portions below the x-axis separately.