If #int_0^4f(x)dx=10# and #int_0^4x*f'(x)dx=2# What is the answer of => f(4)=?

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Answer 1 · 2018-04-09T10:03:16

# 3#.

Recall the Rule of Integration by Parts for Definite Integral :

#int_a^b(u*v')dx=[uv]_a^b-int_a^b(u'*v)dx............(ast)#.

Taking, #a=0, b=4, and, #

#u=x, and, v=f(x)," we have, "u'=1, and, v'=f'(x)#.

#:. (ast) rArr int_0^4xf'(x)dx=[xf(x)]_0^4-int_0^4f(x)dx, or, #

#int_0^4xf'(x)dx=4f(4)-0f(0)-int_0^4f(x)dx#.

Inserting the given values, we get,

#2=4f(4)-0-10 rArr 2+10=4f(4) rArr f(4)=3#.

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