If f and g are defined as f(x) = f (a -x) and g(x) + g (a -x) = 4, then ∫0a f(x)g(x)dx is equal to

# If f and g are defined as f(x) = f (a -x) and g(x) + g (a -x) = 4, then ${\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

1. A

${\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$

2. B

$2{\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$

3. C

$2{\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$

4. D

None of these

Fill Out the Form for Expert Academic Guidance!l

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.

### Solution:

Let,

$\begin{array}{l}\mathrm{I}={\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}----\left(\mathrm{i}\right)\\ \mathrm{I}={\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{a}-\mathrm{x}\right)\mathrm{g}\left(\mathrm{a}-\mathrm{x}\right)\mathrm{dx}\\ \left[\because {\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{a}-\mathrm{x}\right)\mathrm{dx}\right]\end{array}$

On adding Eqs. (i) and (ir), we get

$2\mathrm{I}={\int }_{0}^{\mathrm{a}} 4\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}⇒\mathrm{I}=2{\int }_{0}^{\mathrm{a}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$

## Related content

 NEET 2024 Application Form: Your Complete Guide to Avoiding Costly Mistakes Guide to NEET Correction Window 2024 – Step-by-Step Instructions for Perfecting Your NEET Form CBSE Class 6 Maths Important Topics Best Online Science Tuition For Class 6 CBSE Syllabus for Class 2 Hindi Updated 2023-24 CBSE Worksheets for Class 6 Maths CBSE Board Exam 2024 Begin: Check Guidelines And Instructions CBSE Class 11 Economics Syllabus CBSE Syllabus for Class 1 EVS Updated 2023-24 CBSE Class 11 Business Studies Syllabus 2023-24

Talk to our academic expert!

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

I agree to the terms and conditions and privacy policy.