Learn How To Solve Math Problems Using Chain Rule And Fundamental Theorem Of Calculus

ⅆⅆx(∫x2sin(t4)ⅆt)=

sin(x4) – x to the fourth

To solve this problem, we need to apply the chain rule and the fundamental theorem of calculus.

Firstly, let’s simplify the integrand with respect to t using substitution method.

Let u = t^4

Then, du/dt = 4t^3

and dt = du/4t^3

Substituting in the integral:

∫x^2 sin(t^4) dt = (∫sin(u)du)/4t^3

= (-cos(u))/4t^3

= (-cos(t^4))/4t^3

Now, let’s apply the chain rule to differentiate the expression with respect to x:

ⅆⅆx(∫x^2 sin(t^4) dt)

= ∫ⅆⅆx (-cos(t^4))/4t^3 ⅆt

= ∫(-cos(t^4))/4t^3 (ⅆt/ⅆx) ⅆt

= ∫(-cos(t^4))/4t^3 (ⅆ(t^4)/ⅆx) ⅆt

= ∫-4x^3 cos(t^4)/4t^3 ⅆt

= -x^3 sin(t^4) + C

Hence, the final result is -x^3 sin(t^4) + C.

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