MHB Derivative of ∫ (1+v^3)^10 dv from sinx to cosx | Ashleigh N.

AI Thread Summary
The discussion focuses on finding the derivative of the integral from sin(x) to cos(x) of the function (1+v^3)^10. The approach involves applying the Fundamental Theorem of Calculus (FTOC) and the chain rule. The derivative is expressed as the difference between the derivatives of the function F evaluated at cos(x) and sin(x). The final result is given as a combination of terms involving sine and cosine functions, specifically: -sin(x)(1+cos^3(x))^10 - cos(x)(1+sin^3(x))^10. This method effectively demonstrates the application of calculus principles to evaluate the derivative of a definite integral with variable limits.
MarkFL
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Here is the question:

Find the derivative of the integral from sinx to cosx of (1+v^3)^10?

cosx
y = ∫ (1+v^3)^10 dv
sinx

I have posted a link there to this thread so the OP can view my work.
 
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Hello Ashleigh N.,

We are given to evaluate:

$$\frac{d}{dx}\left(\int_{\sin(x)}^{\cos(x)} \left(1+v^3 \right)^{10}\,dv \right)$$

If we define $F(u)$ to be a function such that:

$$\frac{dF}{du}=f(u)=\left(1+u^3 \right)^{10}$$ then by the FTOC and the chain rule, we may state:

$$\frac{d}{dx}\left(\int_{\sin(x)}^{\cos(x)} \left(1+v^3 \right)^{10}\,dv \right)=\frac{d}{dx}\left(F\left(\cos(x) \right)-F\left(\sin(x) \right) \right)=\frac{d}{dx}F\left(\cos(x) \right)-\frac{d}{dx}F\left(\sin(x) \right)=$$

$$f\left(\cos(x) \right)\frac{d}{dx}\cos(x)-f\left(\sin(x) \right)\frac{d}{dx}\sin(x)=-\sin(x)\left(1+\cos^3(x) \right)^{10}-\cos(x)\left(1+\sin^3(x) \right)^{10}$$
 
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