Question

Assertion :${\int }_{-1}^1\frac{\sin x-x^4}{4-\left|x\right|}dx$ is same as ${\int }_0^1\frac{-2x^4}{4-\left|x\right|}dx$ Reason:

${\int }_{-1}^1(f\left(x\right)+g\left(x\right))dx=2{\int }_0^{1}f\left(x\right)dx$ if $g\left(x\right)$ is an odd function and $f\left(x\right)$ is an even function.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

${\int }_{-1}^1\frac{\sin x-x^4}{4-\left|x\right|}dx$
$={\int }_{-1}^1\frac{\sin x}{4-\left|x\right|}dx+{\int }_{-1}^1\frac{-x^4}{4-\left|x\right|}dx={\int }_{-1}^{1}g\left(x\right)dx+{\int }_{-1}^{1}f\left(x\right)dx$
Where $g\left(x\right)=\frac{\sin x}{4-\left|x\right|}$ & $f\left(x\right)=\frac{-x^4}{4-\left|x\right|}$
$=2{\int }_0^{1}f\left(x\right)dx$
As $f(-x)=f\left(x\right)$ i.e. an even function and $g\left(x\right)$ is an odd function.
$=2{\int }_0^1\frac{-x^4}{4-\left|x\right|}dx$
Hence Reason (R) is solution for Assertion (A).