If f is continuous and integral 0 to 4 $f\left(x\right)dx=10$ , find integral 0 to 2 $f\left(2x\right)dx$

Cabiolab
2021-10-07
Answered

If f is continuous and integral 0 to 4 $f\left(x\right)dx=10$ , find integral 0 to 2 $f\left(2x\right)dx$

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StrycharzT

Answered 2021-10-08
Author has **102** answers

We will start working with the integral ${\int}_{0}^{2}f\left(2x\right)dx$

Substitute$2x=u$ and $2dx=du$

$=\frac{1}{2}{\int}_{0}^{2}f\left(2x\right)\left(2dx\right)$

Limits of integration will change from$\int}_{0}^{2$ to $\int}_{2\times 0}^{2\times 2}={\int}_{0}^{4$

$\frac{1}{2}{\int}_{0}^{4}f\left(u\right)du$

Remember that${\int}_{a}^{b}f\left(u\right)du={\int}_{a}^{b}f\left(x\right)dx$

$=\frac{1}{2}{\int}_{0}^{4}f\left(x\right)dx$

It is given that

${\int}_{0}^{4}f\left(x\right)dx=10$

$=\frac{1}{2}\times 10=5$

Result:${\int}_{0}^{2}f\left(2x\right)dx=5$

Substitute

Limits of integration will change from

Remember that

It is given that

Result:

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