We have already seen how to find the antiderivative of some very simple polynomials
For example, the antiderivative of a constant term $a$a, is $ax+C$ax+C
We can also see some examples of powers of $x$x,
Which leads to a general form of
To find indefinite integrals of polynomial functions we combine the idea that the antiderivative of a sum is the same as the sum of the anitderivatives and our rule for finding the antiderivative of a power term together.
Find $$
$\int3x^4+4x^3-2x^2+x-7dx$∫3x4+4x3−2x2+x−7dx | $=$= | $\int3x^4dx+\int4x^3dx-\int2x^2dx+\int xdx-\int7dx$∫3x4dx+∫4x3dx−∫2x2dx+∫xdx−∫7dx |
$=$= | $3\int x^4dx+4\int x^3dx-2\int x^2dx+\int xdx-\int7dx$3∫x4dx+4∫x3dx−2∫x2dx+∫xdx−∫7dx | |
$=$= | $3\times\frac{x^5}{5}+4\times\frac{x^4}{4}-2\times\frac{x^3}{3}+\frac{x^2}{2}-7x+C$3×x55+4×x44−2×x33+x22−7x+C | |
$=$= | $\frac{3x^5}{5}+\frac{4x^4}{4}-\frac{2x^3}{3}+\frac{x^2}{2}-7x+C$3x55+4x44−2x33+x22−7x+C | |
$=$= | $\frac{3x^5}{5}+x^4-\frac{2x^3}{3}+\frac{x^2}{2}-7x$3x55+x4−2x33+x22−7x |
A particular curve has a derivative given by $\frac{dy}{dx}=2x^4-3x^2+x$dydx=2x4−3x2+x
We are told that the curve passes through the point $(1,-\frac{1}{2})$(1,−12). Find the equation of the curve.
We start here with the indefinite integral
$\int2x^4-3x^2+xdx$∫2x4−3x2+xdx | $=$= | $2\int x^4dx-3\int x^2dx+\int xdx$2∫x4dx−3∫x2dx+∫xdx |
$=$= | $2\cdot\frac{x^5}{5}-3\cdot\frac{x^3}{3}+\frac{x^2}{2}+C$2·x55−3·x33+x22+C | |
$=$= | $\frac{2x^5}{5}-x^3+\frac{x^2}{2}+C$2x55−x3+x22+C |
We are given a set of conditions that can help us identify the value of $C$C. We use the point $(1,-\frac{1}{2})$(1,−12)
$y$y | $=$= | $\frac{2x^5}{5}-x^3+\frac{x^2}{2}+C$2x55−x3+x22+C |
$-\frac{1}{2}$−12 | $=$= | $\frac{2}{5}-1+\frac{1}{2}+C$25−1+12+C |
$-\frac{1}{2}-\frac{2}{5}+1-\frac{1}{2}$−12−25+1−12 | $=$= | $C$C |
$C$C | $=$= | $-\frac{2}{5}$−25 |
So the curve is $y=\frac{2x^5}{5}-x^3+\frac{x^2}{2}-\frac{2}{5}$y=2x55−x3+x22−25
Find the primitive function of $10x+7$10x+7.
Use $C$C as the constant of integration.
Find the primitive function of $15x^4+16x^3$15x4+16x3.
Use $C$C as the constant of integration.
Find an equation for $y$y if $\frac{dy}{dx}=\left(5x-2\right)\left(3x-4\right)$dydx=(5x−2)(3x−4).
Use $C$C as the constant of integration.
Apply differentiation and anti-differentiation techniques to polynomials
Apply calculus methods in solving problems