If $f(x)=\cos x-\int\limits_0^t(x-t)f(t)

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Target Exam

CUET

Subject

-- Mathematics - Section A

Chapter

Definite Integration

Question:

If $f(x)=\cos x-\int\limits_0^t(x-t)f(t)dt$, then f''(x) + f(x) equals:

Options:

-cos x

$\int_0^t(x-t)f(t)dt$

$-\int_0^t(x-t)f(t)dt$

Correct Answer:

$-\int_0^t(x-t)f(t)dt$

Explanation:

$f(x)=\cos x-\int\limits_0^t(x-t)f(t)dt⇒f'(x)=-\sin x-\int\limits_0^t\frac{d}{dx}[(x-t)f(t)]dt$

$⇒f'(x)=-\sin x-\int\limits_0^t[f(t)-0]dt⇒f''(x)=-\cos x - 0$

$⇒f(x)+f''(x)=\cos x-\int\limits_0^t(x-t)f(t)dt+(-\cos x)$