Question

Find \(\frac{d}{d x} \int_{3-x}^{x^{2}}(x-t) d t\) by evaluating the integral first, and by differentiating first.

Short answer

The derivative of the given integral is \(2x - x^4 - 6 + x^2\).

Step-by-step solution

Understand the problem

The problem requires finding the derivative of the integral \(\frac{d}{dx} \int_{3-x}^{x^2} (x-t) \, dt\). This can be approached in two ways: evaluating the integral first and then differentiating, or differentiating the integral directly considering Leibniz's rule for differentiation under the integral sign.

Evaluate the integral first

First, let's evaluate \(\int_{3-x}^{x^2} (x-t) \, dt\).Integrate (x-t) with respect to t:\(\int (x-t) \, dt = xt - \frac{t^2}{2} + C\)Now apply the limits from \(3-x\) to \(x^2\):\[ \left[ xt - \frac{t^2}{2} \right]_{3-x}^{x^2} = \left( x(x^2) - \frac{(x^2)^2}{2} \right) - \left( x(3-x) - \frac{(3-x)^2}{2} \right) \]Simplify the expression.

Simplify the evaluated integral

Simplify each part:For the upper limit \(x^2\):\[ x(x^2) - \frac{(x^2)^2}{2} = x^3 - \frac{x^4}{2} \]For the lower limit \(3-x\):\[ x(3-x) - \frac{(3-x)^2}{2} = 3x - x^2 - \frac{(9-6x+x^2)}{2} = 3x - x^2 - \frac{9}{2} + 3x - \frac{x^2}{2} \]Combine:\[ x^3 - \frac{x^4}{2} - (3x - x^2 - \frac{9}{2} + 3x - \frac{x^2}{2}) = x^3 - \frac{x^4}{2} - 6x + x^2 + 1.5 + \frac{x^2}{2} \]Simplify to get the final integral expression.

Differentiate the simplified integral

Differentiate the simplified expression with respect to x.Use the result from Step 3, denoted as f(x):\( f(x) = x^3 - \frac{x^4}{2} - 6x + x^2 + \frac{3}{2} + \frac{x^2}{2} \)Differentiate \(f(x)\):\[ \frac{d}{d x}f(x) = 3x^2 - 2x^3 - 6 + 2x + x \]

Differentiating first using Leibniz's Rule

Apply Leibniz's Rule to differentiate under the integral sign:\( \frac{d}{dx} \int_{3-x}^{x^2} (x-t) \, dt = (x - t) \bigg|_{t = x^2} \frac{d}{dx} (x^2) - (x - t) \big|_{t = 3-x} \frac{d}{dx} (3-x) + \int_{3-x}^{x^2} \frac{d}{dx}(x - t) \, dt \)This expression evaluates to:For the first term:\[ (x - x^2) \cdot 2x \]For the second term:\[ (x - (3-x)) \cdot (-1) \]For the third term:\[ \int_{3-x}^{x^2} 1 \, dt \]Combine these.

Simplify the differentiated integral

Combine the expressions from the Leibniz’s Rule application:\( (x - x^2) \cdot 2x + (2x - 3) \cdot (-1) + \int_{3-x}^{x^2} 1 \, dt \)Simplify each term:

Finalize the result

Evaluate and simplify the entire expression to finalize the answer.

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics focused on integrals and their properties.
It allows us to determine quantities like area under a curve and accumulation of quantities.
In this problem, we start with an integral that has variable limits and an integrand, both dependent on the variable of differentiation.

To evaluate the integral \(\frac{d}{dx} \int_{3-x}^{x^2} (x-t) \ dt\), we'll simplify it before differentiating or use integration techniques like substitution.

Leibniz's Rule

Named after the mathematician Gottfried Wilhelm Leibniz, Leibniz's Rule helps us differentiate an integral with variable limits of integration.

The rule states that if you have an integral dependent on another variable, you can take the derivative by considering both the change in the limits and the integrand.

Formally, for an integral \(\frac{d}{d x} \int_{a(x)}^{b(x)} f(x,t) \ dt\):
\[ \frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) \, dt = f(x,b(x)) \cdot b'(x) - f(x,a(x)) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) \, dt \]
This rule simplifies our exercise: it breaks down the differentiation of the integral into more manageable parts.
We follow each part separately using products and contributions from the limits.

Definite Integrals

A definite integral calculates the area between the curve and the x-axis within given limits.

In this exercise, we evaluate the definite integral from \(3-x\) to \(x^2\).
After determining the antiderivative \(xt - \frac{t^2}{2} + C\), we apply these limits:
\[\int_{3-x}^{x^2} (x-t) \, dt = \left[ xt - \frac{t^2}{2} \right]_{3-x}^{x^2} \]
By substituting the upper and lower limits, we simplify the integral.
Upper limit \(x^2\): \(x(x^2) - \frac{(x^2)^2}{2} = x^3 - \frac{x^4}{2}\)
Lower limit \(3-x\): \(x(3-x) - \frac{(3-x)^2}{2}\).
Combining and simplifying these results provides a clear integral expression ready for differentiation.

Derivatives

Derivatives measure how functions change as their inputs change.
They are the foundation of differential calculus.

In our exercise, after simplifying the integral, we differentiate the resulting function:
\[ f(x) = x^3 - \frac{x^4}{2} - 6x + x^2 + \frac{3}{2} + \frac{x^2}{2} \]
To find \( \frac{d f(x)}{d x} \):
- Derivative of \(x^3\) is \(3x^2\)
- Derivative of \( - \frac{x^4}{2} \) is \(-2x^3\)
- Derivative of \(6x\) is \(-6\)
- Derivative of \( x^2\) is \(2x\)
- The derivative of constants like \( \frac{3}{2} \) is zero.

Combining these results, we derive the final simplified expression.
This procedure shows how integral and differential calculus are connected, offering a comprehensive problem solution.