How Do You Find The Integral T Sint Dt ? Integral Of Tsint

I need to find the derivative of this function. I know I need to separate the integrals into two and use the chain rule but I am stuck.

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$$y=int_sqrt{x}^{x^3}sqrt{t}sin t~dt~.$$

Thanks in advance



Let me show you a general method which works in these sorts of situations.

By the Fundamental Theorem of Calculus, we know how to take the derivative of$$F(z):=int_0^zsqrt{t}sin(t),dt;$$in particular, FTC tells us that $$ ag{1}F”(z)=sqrt{z}sin(z).$$Now, note that$$int_{sqrt{x}}^{x^3}sqrt{t}sin(t),dt=int_0^{x^3}sqrt{t}sin(t),dt-int_0^{sqrt{x}}sqrt{t}sin(t),dt=F(x^3)-F(sqrt{x}).$$So, the derivative you want is$$frac{d}{dx}left.$$See if you can use the Chain Rule, and (1), to finish it up from here.

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By the chain rule we prove easly:

If $$F(x)=int_{u(x)}^{v(x)}f(t)dt$$then$$F”(x)=f(v(x))v”(x)-f(u(x))u”(x)$$

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