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.

Question

BlackTom AI · Accepted Answer

This problem asks you to apply the fundamental theorem of calculus to a definite integral with a variable upper limit.

Recall the fundamental theorem of calculus (Part I): if F(x) = ∫ from a to x of f(t) dt, then F'(x) = f(x), provided f is continuous on the interval.

Here, f(t) = t cos t and the lower limit a = 0, so the derivative with respect to x of ∫ from 0 to x t cos t dt is simply f(x) = x cos x.

Therefore, the derivative is x cos x, which matches the provided option.

Note: there are no other answer choices listed to compare, so there isn’t an alternative option to evaluate in this particular item. The key idea is recognizing that the derivative of an integral with a variable upper limit evaluates to the integrand at that upper limit.

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.

View Explanation

Log in for full answers

Similar Questions

Find the derivative.

Let [math: F(x)=∫03xet2 dt]\displaystyle F(x)=\int _0^{3x}{e^{t^2} dt}. Compute [math: F′(0.5)]F’(0.5). (Correct the answer to 2 decimal places.)

Let \(f(x) = e^{\sin^2(2\pi x)}\). Calculate $$\int_{0}^{1} f'(x) \, dx $$

Use the fundamental theorem of calculus to find the following derivative:\dfrac{d}{dx} \displaystyle \int_0^{x^2} \dfrac{1}{\sqrt{1+t^2}}\, dt.Choose the correct response from below.

If 𝑓 ( 4 ) = 8 , 𝑓 ′ ( 𝑥 ) is continuous, and ∫ 4 10 𝑓 ′ ( 𝑥 ) 𝑑 𝑥 = − 1 , what is 𝑓 ( 10 ) ?

Given that 𝐺 ( 𝑥 ) = ∫ 3 𝑥 cos ⁡ ( 𝜋 𝑡 / 6 ) 𝑡 2 𝑑 𝑡 , what is 𝐺 ′ ( 2 ) ?

Use the fundamental theorem of calculus to find the following derivative:\dfrac{d}{dx} \displaystyle \int_0^{x^2} \dfrac{1}{\sqrt{1+t^2}}\, dt.Choose the correct response from below.

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.

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler