Find the integral from the image given below

Question

Equation problem click here

$$\frac{d}{dx} \int_{x^4}^{x^3}\cos\left(t^3+t\right)\space{dt}= \;?$$

Does anyone know how to solve this problem??

Answer 1

In general:

$$\frac{\text{d}}{\text{d}x}\left\{\int_{\text{g}\left(x\right)}^{\text{y}\left(x\right)}\text{f}\left(t\right)\space\text{d}t\right\}=\text{f}\left(\text{y}\left(x\right)\right)\cdot\text{y}\space'\left(x\right)-\text{f}\left(\text{g}\left(x\right)\right)\cdot\text{g}\space'\left(x\right)\tag1$$

So, in your problem:

$$\frac{\text{d}}{\text{d}x}\left\{\int_{x^4}^{x^3}\cos\left(t^3+t\right)\space\text{d}t\right\}=$$ $$\cos\left(\left(x^3\right)^3+x^3\right)\cdot\frac{\text{d}}{\text{d}x}\left(x^3\right)-\cos\left(\left(x^4\right)^3+x^4\right)\cdot\frac{\text{d}}{\text{d}x}\left(x^4\right)=$$ $$3x^2\cos\left(x^9+x^3\right)-4x^3\cos\left(x^{12}+x^4\right)\tag1$$