/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 If \(F(t)=\int_{2}^{3} \sin \lef... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 1

If \(F(t)=\int_{2}^{3} \sin \left(x+t^{2}\right) d x\), find \(F^{\prime}(t)\).

Short Answer

Expert verified
Answer: The derivative of the function \(F(t)\) with respect to \(t\) is \(F'(t) = \int_{2}^{3} \cos(x + t^2) \cdot 2t dx\).

Step by step solution

01

Find the partial derivative of \(f(x, t)\)

First, we need to find the partial derivative of \(f(x, t) = \sin(x + t^2)\) with respect to \(t\). Using the chain rule, we get: $$ \frac{\partial}{\partial t} f(x, t) = \frac{\partial}{\partial t} \sin(x + t^2) = \cos(x + t^2) \cdot \frac{\partial}{\partial t}(x + t^2) $$ Now, we find the derivative of \((x + t^2)\) with respect to \(t\): $$ \frac{\partial}{\partial t} (x + t^2) = 2t $$ Therefore, $$ \frac{\partial}{\partial t} f(x, t) = \cos(x + t^2) \cdot 2t $$
02

Integrate the partial derivative with respect to \(x\)

Now that we have the partial derivative, we can use the Leibniz rule to find \(F'(t)\): $$ F'(t) = \int_{2}^{3} \frac{\partial}{\partial t} f(x, t) dx = \int_{2}^{3} \cos(x + t^2) \cdot 2t dx $$
03

Write the final answer

We have found the derivative \(F'(t)\): $$ F'(t) = \int_{2}^{3} \cos(x + t^2) \cdot 2t dx $$ This is our final answer for the derivative of \(F(t)\).

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Most popular questions from this chapter

Evaluate the integrals (i) \(\int_{0}^{b} \frac{x d x}{(1+x)^{3}}\) (ii) \(\int_{0}^{b} \frac{x^{2} d x}{(1+x)^{4}}\) and show that they converge to finite limits as \(\mathrm{b} \rightarrow \infty\)

Compute (a) \(\lim _{t \rightarrow 0+} \int_{t}^{1} \frac{1}{x} \mathrm{dx}\) (b) \(\lim _{t \rightarrow 1-} \int_{0}^{t} \tan \frac{\pi}{2} x d x\). How does the result give insight into the fact that neither integrand is integrable over the interval \([0,1] ?\)

One of the numbers \(\pi, \pi / 2,35 \pi / 128,1-\pi\) is the correct value of the integral \(\int_{0}^{\pi} \sin ^{8} x d x\). Use the graph of \(\mathrm{y}=\sin ^{8} \mathrm{x}\) and a logical process of elimination to find the correct value.

Explain why each of the following integrals is improper. (a) \(\int_{1}^{\infty} x^{4} e^{-x^{4}} d x\) (b) \(\int_{0}^{\pi / 2} \sec x d x\) (c) \(\int_{0}^{2} \frac{x}{x^{2}-5 x+6} d x\) (d) \(\int_{-\infty}^{0} \frac{1}{x^{2}+5} d x\)

Prove that (a) \(\pi=\lim _{n \rightarrow \infty} \frac{4}{n^{2}}\left(\sqrt{n^{2}-1}+\sqrt{n^{2}-2^{2}}+\ldots+\sqrt{n^{2}-n^{2}}\right)\). (b) \(\int_{1}^{3}\left(x^{2}+1\right) d x=\lim _{n \rightarrow \infty} \frac{4}{n^{3}} \sum_{i=1}^{n}\left(n^{2}+2 i n+2 i^{2}\right)\).
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