Given-a-function-f-x-3-f-x-for-x-R-If-3-6-f-x-dx-6-then-3-9-f-x-dx-

Question Number 109101 by bemath last updated on 21/Aug/20

G i v e n a f u n c t i o n f (x + 3) = f (x)

f o r \forall x \in R . I f \underset{- 3}{\int^{6}} f (x) d x = - 6

t h e n \underset{3}{\int^{9}} f (x) d x = ?

Answered by bemath last updated on 21/Aug/20

Answered by john santu last updated on 21/Aug/20

\Rightarrow \underset{3}{\int^{9}} f (x + 3) d x = \underset{3 - 3}{\int^{9 - 3}} f (x + 3 - 3) d (x - 3)

= \underset{0}{\int^{6}} f (x) d x

g i v e n \underset{- 3}{\int^{6}} f (x) d x = \underset{- 3}{\int^{6}} f (x + 3) d x = - 6

l e t x + 3 = u \to {\begin{cases} x = - 3 \Rightarrow u = 0 \\ x = 6 \Rightarrow u = 9 \end{cases}

\underset{0}{\int^{9}} f (u) d u = - 6

s o \underset{0}{\int^{6}} f (x) d x + \underset{6}{\int^{9}} f (x) d x = - 6

w e n e e d i n f o r m a t i o n \underset{6}{\int^{9}} f (x) d x

Answered by mr W last updated on 21/Aug/20

\int_{- 3}^{6} f (x) d x = 3 \int_{0}^{3} f (x) d x = - 6

\Rightarrow \int_{0}^{3} f (x) d x = - 2

\underset{3}{\int^{9}} f (x) d x = 2 \int_{0}^{3} f (x) d x = - 4

Commented by bemath last updated on 21/Aug/20

y e s \dots h o o r r a a y y

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