We are asked to find an antiderivative of `(-2x+5)^6 ` :
This is essentially equivalent to evaluating the indefinite integral `int (-2x+5)^6 `
Now `int u^n du=1/(n+1) u^(n+1) ` with u a differentiable function of x. Here we can use u-substitution with u=-2x+5 and du=-2dx. Multiplying the integrand by -2 and (-1/2) (thus multiplying by 1) we get:
`int (-1/2)(-2)(-2x+5)^6 dx ` or ` -1/2 int (-2x+5)^6(-2dx) `
Integrating we get `(-1/2)(1/7)(-2x+5)^7+C ` where C is some real constant.
Thus an antiderivative for `(-2x+5)^6 ` is `-1/14 (-2x+5)^7 ` .
** We can check by taking the derivative, using the chain rule to get:
`d/(dx)[ -1/14(-2x+5)^7]=(7)(-1/14)(-2x+5)^6(-2)=(-2x+5)^6 ` as required.
** Note that we find an antiderivative, as a whole family of functions exist whose derivative is the given function. We can add any constant to the given antiderivative to get another antiderivative.
** An alternative to integrating is to use guess and revise. We might guess that the function we seek is `(-2x+5)^7 ` . Upon checking by taking the derivative, we note that we are off by a factor of -14 (since the derivative is `7(-2x+5)^6(-2)=-14(-2x+5)^6 ` ), so we introduce the factor -1/14 to compensate.