Use the fundamental theorem of calculus to find the following derivative:\dfrac{d}{dx} \displaystyle \int_0^x t\cos(t)\, dt.Choose the correct response from below.

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The problem asks us to differentiate an integral with a variable upper limit. According to the fundamental theorem of calculus part 1, if F(x) = ∫ from a to x of f(t) dt, then F'(x) = f(x). Here, f(t) = t cos t and a = 0, so d/dx ∫_0^x t cos(t) dt = (t cos t) evaluated at t = x, which is x cos x.

A key point to remember is that the derivative with respect to the upper limit x simply feeds x into the integrand: you don’t differentiate the integrand with respect to x inside the integral, you replace t by x in the integrand once the upper limit is differentiated away.

If you want an extra check, you could perform integration by parts on ∫ t cos t dt to obtain an antiderivative, then differentiate that antiderivative with respect to x; you should recover x cos x as well, consistent with the FTC result.

Since the answer options are not provided here, note that the expression x cos x is the outcome of applying the fundamental theorem of calculus to this specific setup.

Use the fundamental theorem of calculus to find the following derivative:\dfrac{d}{dx} \displaystyle \int_0^x t\cos(t)\, dt.Choose the correct response from below.

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