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Math  /  Calculus

QuestionCompute: ddx∫xx3tsin⁡(2t)dt\frac{d}{d x} \int_{\sqrt{x}}^{x^{3}} t \sin (2 t) d t

Studdy Solution

STEP 1

1. We are dealing with a function defined by an integral with variable limits.
2. The function inside the integral is tsin⁡(2t) t \sin(2t) .
3. We need to find the derivative of this integral with respect to x x .

STEP 2

1. Apply the Leibniz rule for differentiating under the integral sign.
2. Differentiate the upper limit of the integral.
3. Differentiate the lower limit of the integral.
4. Combine the results from steps 2 and 3 to find the derivative.

STEP 3

Apply the Leibniz rule for differentiation under the integral sign:
The Leibniz rule states that if F(x)=∫a(x)b(x)f(t) dt F(x) = \int_{a(x)}^{b(x)} f(t) \, dt , then:
ddxF(x)=f(b(x))⋅b′(x)−f(a(x))⋅a′(x)\frac{d}{dx} F(x) = f(b(x)) \cdot b'(x) - f(a(x)) \cdot a'(x)
where f(t)=tsin⁡(2t) f(t) = t \sin(2t) , a(x)=x a(x) = \sqrt{x} , and b(x)=x3 b(x) = x^3 .

STEP 4

Differentiate the upper limit of the integral:
1. Evaluate f(t) f(t) at t=b(x)=x3 t = b(x) = x^3 :
f(x3)=x3sin⁡(2x3)f(x^3) = x^3 \sin(2x^3)
2. Differentiate the upper limit b(x)=x3 b(x) = x^3 :
b′(x)=ddx(x3)=3x2b'(x) = \frac{d}{dx}(x^3) = 3x^2

STEP 5

Differentiate the lower limit of the integral:
1. Evaluate f(t) f(t) at t=a(x)=x t = a(x) = \sqrt{x} :
f(x)=xsin⁡(2x)f(\sqrt{x}) = \sqrt{x} \sin(2\sqrt{x})
2. Differentiate the lower limit a(x)=x a(x) = \sqrt{x} :
a′(x)=ddx(x)=12xa'(x) = \frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}

STEP 6

Combine the results from steps 2 and 3:
ddx∫xx3tsin⁡(2t) dt=f(x3)⋅b′(x)−f(x)⋅a′(x)\frac{d}{dx} \int_{\sqrt{x}}^{x^3} t \sin(2t) \, dt = f(x^3) \cdot b'(x) - f(\sqrt{x}) \cdot a'(x)
Substitute the expressions found:
=(x3sin⁡(2x3))⋅(3x2)−(xsin⁡(2x))⋅(12x)= \left( x^3 \sin(2x^3) \right) \cdot (3x^2) - \left( \sqrt{x} \sin(2\sqrt{x}) \right) \cdot \left( \frac{1}{2\sqrt{x}} \right)
Simplify the expression:
=3x5sin⁡(2x3)−12sin⁡(2x)= 3x^5 \sin(2x^3) - \frac{1}{2} \sin(2\sqrt{x})
The derivative of the integral is:
3x5sin⁡(2x3)−12sin⁡(2x)\boxed{3x^5 \sin(2x^3) - \frac{1}{2} \sin(2\sqrt{x})}

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