Evaluate The Derivative:$\[ \frac{d}{d X} \int_{\sin X}^4 \sqrt{1+t^2} \, D T \\]

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Introduction

In calculus, the derivative of a definite integral is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. The derivative of a definite integral is used to find the rate of change of the area under a curve with respect to a parameter. In this article, we will evaluate the derivative of a given definite integral using the fundamental theorem of calculus.

The Fundamental Theorem of Calculus

The fundamental theorem of calculus states that if f(x) is a continuous function on the interval [a, b], then the definite integral of f(x) from a to b is equal to the antiderivative of f(x) evaluated at b minus the antiderivative of f(x) evaluated at a. Mathematically, this can be expressed as:

∫abf(x) dx=F(b)−F(a)\int_{a}^{b} f(x) \, dx = F(b) - F(a)

where F(x) is the antiderivative of f(x).

Evaluating the Derivative of a Definite Integral

To evaluate the derivative of a definite integral, we can use the fundamental theorem of calculus. Let's consider the given definite integral:

ddx∫sin⁡x41+t2 dt\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t

Using the fundamental theorem of calculus, we can rewrite the derivative of the definite integral as:

ddx∫sin⁡x41+t2 dt=ddx[F(4)−F(sin⁡x)]\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t = \frac{d}{d x} \left[ F(4) - F(\sin x) \right]

where F(t) is the antiderivative of 1+t2\sqrt{1+t^2}.

Finding the Antiderivative

To find the antiderivative of 1+t2\sqrt{1+t^2}, we can use the following formula:

∫1+t2 dt=12ln⁡∣t+t2+1∣+C\int \sqrt{1+t^2} \, dt = \frac{1}{2} \ln \left| t + \sqrt{t^2 + 1} \right| + C

where C is the constant of integration.

Evaluating the Derivative

Now that we have found the antiderivative of 1+t2\sqrt{1+t^2}, we can evaluate the derivative of the definite integral:

ddx∫sin⁡x41+t2 dt=ddx[F(4)−F(sin⁡x)]\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t = \frac{d}{d x} \left[ F(4) - F(\sin x) \right]

=ddx[12ln⁡∣4+42+1∣−12ln⁡∣sin⁡x+sin⁡2x+1∣]= \frac{d}{d x} \left[ \frac{1}{2} \ln \left| 4 + \sqrt{4^2 + 1} \right| - \frac{1}{2} \ln \left| \sin x + \sqrt{\sin^2 x + 1} \right| \right]

=1214+42+1ddx[4+42+1]−121sin⁡x+sin⁡2x+1ddx[sin⁡x+sin⁡2x+1]= \frac{1}{2} \frac{1}{4 + \sqrt{4^2 + 1}} \frac{d}{d x} \left[ 4 + \sqrt{4^2 + 1} \right] - \frac{1}{2} \frac{1}{\sin x + \sqrt{\sin^2 x + 1}} \frac{d}{d x} \left[ \sin x + \sqrt{\sin^2 x + 1} \right]

Simplifying the Expression

To simplify the expression, we can use the chain rule and the fact that the derivative of sin⁡x\sin x is cos⁡x\cos x:

ddx[4+42+1]=0\frac{d}{d x} \left[ 4 + \sqrt{4^2 + 1} \right] = 0

ddx[sin⁡x+sin⁡2x+1]=cos⁡x+cos⁡xsin⁡2x+1\frac{d}{d x} \left[ \sin x + \sqrt{\sin^2 x + 1} \right] = \cos x + \frac{\cos x}{\sqrt{\sin^2 x + 1}}

Final Answer

Substituting the simplified expressions back into the original equation, we get:

ddx∫sin⁡x41+t2 dt=1214+42+1⋅0−121sin⁡x+sin⁡2x+1[cos⁡x+cos⁡xsin⁡2x+1]\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t = \frac{1}{2} \frac{1}{4 + \sqrt{4^2 + 1}} \cdot 0 - \frac{1}{2} \frac{1}{\sin x + \sqrt{\sin^2 x + 1}} \left[ \cos x + \frac{\cos x}{\sqrt{\sin^2 x + 1}} \right]

=−cos⁡x2[sin⁡x+sin⁡2x+1]= -\frac{\cos x}{2 \left[ \sin x + \sqrt{\sin^2 x + 1} \right]}

Conclusion

In this article, we evaluated the derivative of a given definite integral using the fundamental theorem of calculus. We found the antiderivative of 1+t2\sqrt{1+t^2} and then used the chain rule and the fact that the derivative of sin⁡x\sin x is cos⁡x\cos x to simplify the expression. The final answer is:

ddx∫sin⁡x41+t2 dt=−cos⁡x2[sin⁡x+sin⁡2x+1]\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t = -\frac{\cos x}{2 \left[ \sin x + \sqrt{\sin^2 x + 1} \right]}

This result has numerous applications in various fields, including physics, engineering, and economics.

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Introduction

In our previous article, we evaluated the derivative of a given definite integral using the fundamental theorem of calculus. In this article, we will answer some frequently asked questions related to the topic.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus states that if f(x) is a continuous function on the interval [a, b], then the definite integral of f(x) from a to b is equal to the antiderivative of f(x) evaluated at b minus the antiderivative of f(x) evaluated at a.

Q: How do I find the antiderivative of a function?

A: To find the antiderivative of a function, you can use various techniques such as substitution, integration by parts, and integration by partial fractions. You can also use a calculator or a computer algebra system to find the antiderivative.

Q: What is the chain rule in calculus?

A: The chain rule is a fundamental concept in calculus that states that if you have a composite function, you can find the derivative of the function by multiplying the derivatives of the individual functions.

Q: How do I apply the chain rule to find the derivative of a definite integral?

A: To apply the chain rule to find the derivative of a definite integral, you need to find the antiderivative of the function and then use the chain rule to find the derivative of the antiderivative.

Q: What is the final answer to the problem?

A: The final answer to the problem is:

ddx∫sin⁡x41+t2 dt=−cos⁡x2[sin⁡x+sin⁡2x+1]\frac{d}{d x} \int_{\sin x}^{4} \sqrt{1+t^2} \, d t = -\frac{\cos x}{2 \left[ \sin x + \sqrt{\sin^2 x + 1} \right]}

Q: What are some common applications of the derivative of a definite integral?

A: The derivative of a definite integral has numerous applications in various fields, including physics, engineering, and economics. Some common applications include:

  • Finding the rate of change of the area under a curve
  • Finding the rate of change of the volume of a solid
  • Finding the rate of change of the work done by a force
  • Finding the rate of change of the energy of a system

Q: How do I use the derivative of a definite integral in real-world problems?

A: To use the derivative of a definite integral in real-world problems, you need to identify the function that represents the quantity you are interested in, find the antiderivative of the function, and then use the chain rule to find the derivative of the antiderivative.

Q: What are some common mistakes to avoid when evaluating the derivative of a definite integral?

A: Some common mistakes to avoid when evaluating the derivative of a definite integral include:

  • Failing to find the antiderivative of the function
  • Failing to use the chain rule to find the derivative of the antiderivative
  • Making errors in the calculation of the derivative
  • Failing to check the units of the answer

Conclusion

In this article, we answered some frequently asked questions related to the topic of evaluating the derivative of a definite integral. We hope that this article has been helpful in clarifying any confusion and providing a better understanding of the topic.