For -fx dx-Fx-C- if -4f2x-3 dx-a F2x-3-C- then the value of a is

For any integrable function nbsp;f(x), such that nbsp;∫f(x)dx=F(x)+C We know that ∫kf(x)dx=kF(x)+C and nbsp; ∫f(ax+b)dx=1aF(ax+b)+C Thus combining these t ...

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Byju's Answer

Standard XIII

Mathematics

Theorems on Integration

For ∫fx dx=Fx...

Question

For $\int f (x) d x = F (x) + C$ , if $\int 4 f (2 x + 3) d x = a F (2 x + 3) + C$ , then the value of $a$ is

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Solution

The correct option is A 2
For any integrable function $f (x)$ , such that $\int f (x) d x = F (x) + C$
We know that
$\int k f (x) d x = k F (x) + C$
and
$\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C$
Thus combining these two properties we get:
$\int k f (a x + b) d x = \frac{k}{a} F (a x + b) + C$

Now we can write our integral as:
$\int 4 f (2 x + 3) d x = \frac{4}{2} F (2 x + 3) + C$

,which simplifies to
$\int 4 f (2 x + 3) d x = 2 F (2 x + 3) + C$

By comparing these equations, we get $a = 2$

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For ∫f(x) dx=F(x)+C, if ∫4f(2x+3) dx=a F(2x+3)+C, then the value of a is

For $\int f (x) d x = F (x) + C$ , if $\int 4 f (2 x + 3) d x = a F (2 x + 3) + C$ , then the value of $a$ is