Evaluate the following integrals. ∫81x3−9x2dx

Gregory Jones

Gregory Jones

Answered

2022-01-03

Evaluate the following integrals.
81x39x2dx

Answer & Explanation

Joseph Fair

Joseph Fair

Expert

2022-01-04Added 34 answers

Step 1
Given: I=81x39x2dx
for evaluating given integral, we first simplify it then integrate is
so,
81dxx39x2=81dxx2(x9)
=81[181x19x2+181(x9)]dx
=81[181ln|x|+19x+181ln|x9|]+c
(dxxa=ln|xa|+c,xndx=xn+1n+1+c
=ln|x|+9x+ln|x9|+c)
Step 2
hence, given integral is ln|x9|ln|x|+9x+c.
redhotdevil13l3

redhotdevil13l3

Expert

2022-01-05Added 30 answers

81x39x2dx
811x39x2dx
81181x19x2+181(x9)dx
81(181xdx19x2+181(x9))dx
81(181ln(|x|)+19x+181ln(|x9|))
ln(|x|)+9x+ln(|x9|)
Solution:
ln(|x|)+9x+ln(|x9|)+C
Vasquez

Vasquez

Expert

2022-01-07Added 457 answers

81x39x2dx
Let's represent it in the form:
81x2(x9)=81x2(x9)
We use the method of decomposition into the elementary elements. Let us expand the function into the simplest terms:
81x2(x9)=Ax+Bx2+Cx9=Ax(x9)+B(x9)+Cx2x2(x9)
81=Ax(x9)+B(x9)+Cx2
x2:A+C=0
x:-9A+B=0
1: -9B=81
Solving it, we find:
A=-1; B=-9; C=1
81x2(x9)=1x+9x2+1x9
We calculate the tabular integral: We
1x9dx=ln(x9)
calculate the tabular integral: We
1xdx=ln(x)
calculate the tabular integral:
9x2dx=9x
Answer:
ln(x9)ln(x)+9x+C

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