Math Problem Statement

etermine the following derivatives: (i) d dx (∫ x 2 t sin t dt )

Solution

To find the derivative of the given expression, let's first rewrite it clearly:

Let F(x)=ddx(2xtsintdt)\text{Let } F(x) = \frac{d}{dx} \left(\int_{2}^{x} t \sin t \, dt \right)

To solve this, we can apply the Leibniz rule for differentiation under the integral sign, which is a special case of the Fundamental Theorem of Calculus. The Leibniz rule states:

ddx(a(x)b(x)f(t,x)dt)=f(b(x),x)b(x)f(a(x),x)a(x)+a(x)b(x)xf(t,x)dt\frac{d}{dx} \left( \int_{a(x)}^{b(x)} f(t, x) \, dt \right) = f(b(x), x) \cdot b'(x) - f(a(x), x) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(t, x) \, dt

Here, the integral is from a constant a=2a = 2 to b(x)=xb(x) = x. Therefore, the derivative simplifies as:

F(x)=ddx(2xtsintdt)=tsintt=xddx(x)tsintt=2ddx(2)F(x) = \frac{d}{dx} \left( \int_{2}^{x} t \sin t \, dt \right) = t \sin t \bigg|_{t=x} \cdot \frac{d}{dx}(x) - t \sin t \bigg|_{t=2} \cdot \frac{d}{dx}(2)

Since ddx(x)=1\frac{d}{dx}(x) = 1 and ddx(2)=0\frac{d}{dx}(2) = 0, the derivative is:

F(x)=xsinx0=xsinxF(x) = x \sin x - 0 = x \sin x

Thus, the derivative is:

xsinx\boxed{x \sin x}


Would you like further details or have any questions about the solution?

Here are some related questions you might consider:

  1. What is the derivative of 0xet2dt\int_{0}^{x} e^{t^2} \, dt?
  2. How does the Leibniz rule apply when the limits of integration are both functions of xx?
  3. Can you compute the derivative ddx(0x2cos(t)dt)\frac{d}{dx} \left(\int_{0}^{x^2} \cos(t) \, dt \right)?
  4. What if the integrand involved xx explicitly, like 1xx2sintdt\int_{1}^{x} x^2 \sin t \, dt?
  5. How would you differentiate a(x)b(x)f(t)dt\int_{a(x)}^{b(x)} f(t) \, dt when both limits are functions of xx?

Tip: When applying the Leibniz rule, always identify if the limits of integration are constants or functions of xx, as this affects how the rule is applied.

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Math Problem Analysis

Mathematical Concepts

Integral Calculus
Differentiation
Leibniz Rule

Formulas

Leibniz rule for differentiation under the integral sign

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

College Level