Integration and Accumulation of Change

To make the integral more complex.

To transform the integral into a simpler, more recognizable form.

To eliminate the need for antiderivatives.

To find the derivative of the integrand.

Differentiate the entire integrand.

Look for a part of the integrand that can be simplified with a new variable, often an inner function.

Integrate the entire integrand directly.

Square the integrand.

$(5x+3)^5 + C$

$(5x+3)^5/5 + C$

$(5x+3)^5/25 + C$

$(5x+3)^3 + C$

$\frac{1}{3}(x^2+1)^{3/2} + C$

$\frac{2}{3}(x^2+1)^{3/2} + C$

$\frac{1}{2}(x^2+1)^{3/2} + C$

$(x^2+1)^{3/2} + C$

$\frac{1}{2}ln|x^2+4| + C$

$ln|x^2+4| + C$

$\frac{1}{4}ln|x^2+4| + C$

$2ln|x^2+4| + C$

Always substitute back into the expression before evaluating.

Always change the limits of integration as you work through the problem.

Either change the limits of integration as you work through the problem, or substitute back into the expression before evaluating.

Always use a calculator to evaluate.

$\frac{625}{8}$

$\frac{624}{8}$

$\frac{600}{8}$

$\frac{1}{8}$

$\frac{1}{2}ln(2)$

$ln(2)$

$2ln(2)$

$\frac{1}{4}ln(2)$

$\frac{56\sqrt{5} + 8}{15}$

$\frac{28\sqrt{5} - 8}{15}$

$\frac{56\sqrt{5} - 8}{15}$

$\frac{28\sqrt{5} + 8}{15}$

$\frac{2-\sqrt{2}}{3}$

$\frac{\sqrt{2}-2}{3}$

$\frac{2+\sqrt{2}}{3}$

$\frac{-\sqrt{2}-2}{3}$