SOLUTION: Find an antiderivative for −5 ( −4 x^2−1 )^2

Algebra ->  Inequalities -> SOLUTION: Find an antiderivative for −5 ( −4 x^2−1 )^2       Log On


   

Question 971771: Find an antiderivative for −5 ( −4 x^2−1 )^2
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Find an antiderivative for -5%28-4x%5E2-1%29%5E2 
-5%28-4x%5E2-1%29%5E2%22%22=%22%22-5%2816x%5E4%2B8x%5E2%2B1%29%22%22=%22%22-80x%5E4-40x%5E2-5

Since that has 3 terms, 

Let an antiderivative be = a%2Ax%5En%2Bbx%5Em%2Bcx%5Ep

the derivative of a%2Ax%5En%2Bbx%5Em%2Bcx%5Ep is a%2An%2Ax%5E%28n-1%29%2Bb%2Am%2Ax%5E%28m-1%29%2Bc%2Ap%2Ax%5E%28p-1%29

Therefore a%2An%2Ax%5E%28n-1%29%2Bb%2Am%2Ax%5E%28m-1%29%2Bc%2Ap%2Ax%5E%28p-1%29%22%22=%22%22-80x%5E4-40x%5E2-5

Write the 5 as 5%2Ax%5E0

a%2An%2Ax%5E%28n-1%29%2Bb%2Am%2Ax%5E%28m-1%29%2Bc%2Ap%2Ax%5E%28p-1%29%22%22=%22%22-80x%5E4-40x%5E2-5x%5E0

Set the corresponding exponents equal to each other

n-1=4, m-1=2, p-1=0

Solve for the exponents:

n=5, m=3, p=1

Substituting,

a%2A5%2Ax%5E4%2Bb%2A3%2Ax%5E2%2Bc%2A1%2Ax%5E0%22%22=%22%22-80x%5E4-40x%5E2-5x%5E0

5ax%5E4%2B3bx%5E2%2Bcx%5E0%22%22=%22%22-80x%5E4-40x%5E2-5x%5E0

Set the coefficients of like powers of x equal to each other

5a=-80, 3b=-40, c=-5

a=-16, b=-40%2F3, c=-5

Substituting those and n=5, m=3, p=1

So an antiderivative = a%2Ax%5En%2Bbx%5Em%2Bcx%5Ep%22%22=%22%22-16x%5E5-expr%2840%2F3%29x%5E3-5x%5E1

Answer:  -16x%5E5-expr%2840%2F3%29x%5E3-5x

Edwin