Find the solution of {f}{''}{\left({x}\right)}={8}{x}+ \sin{{x}}

bobbie71G 2021-09-16 Answered

Find the solution of f(x)=8x+sinx

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Expert Answer

Cullen
Answered 2021-09-17 Author has 89 answers

Solve d2f(x)dx2=8x+sin(x):
Integrate both sides with respect to x:
df(x)dx=(8x+sin(x))dx=4x2cos(x)+c1, where c1 is an arbitrary constant.
INTERMEDIATE STEPS:
Take the integral:
(8x+sin(x))dx
Integrate the sum term by term and factor out constants:
=sin(x)dx+8xdx
The integral of sin(x) is cos(x):
=cos(x)+8xdx
The integral of x is x22:
=4x2cos(x)+constant
Integrate both sides with respect to x:
f(x)=(4x2cos(x)+c1)dx=4x33sin(x)+xc1+c2, where c2 is an arbitrary constant.
INTERMEDIATE STEPS:
Take the integral:
(c1+4x2cos(x))dx
Integrate the sum term by term and factor out constants:
=c11dxcos(x)dx+4x2dx
The integral of 1 is x:
=c1xcos(x)dx+4x2dx
The integral of cos(x) is sin(x):
=sin(x)+c1x+4x2dx
The integral of x2 is x33:
Answer:
=c1x+4x33sin(x)+constant

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