Calculate integrals: a) $\frac{d}{dx}\int_{5x-1}^{x^3}x\cdot(3\sin t+2\ln t)dt$ and b) $\frac{d}{dx}\ln(\int_{2}^{\cos x}(2t+\sin t)dt)$

Question

I am struggling with integral questions. Any help with the integration of these two functions? What technique should be used?

Calculate:

a) $\displaystyle \frac{d}{dx}\int_{5x-1}^{x^3}x\cdot(3\sin t+2\ln t)dt$

b) $\displaystyle \frac{d}{dx}\ln\left(\int_{2}^{\cos x}(2t+\sin t)dt\right)$

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score 1 · Answer 1 · answered Feb 06 '20 at 12:54

You need to use Leibniz's Rule: If $$F(x)=\int_{\psi {(x)}}^{\phi {(x)}}g(x,t)dt$$ then $$F'(x)=g(x,\phi {(x)})\phi '(x)-g(x,\psi (x))\psi '(x)+\int_{\psi (x)}^{\phi (x)}\frac{\partial}{\partial x}g(x,t)dt$$ Note that in (b) two of the three expressions on the right are 0.

score 0 · Answer 2 · answered Feb 06 '20 at 12:32

There are two ways. First, you can calculate the integrals. Just split the terms. For example, write the first expression as $$\frac {d}{dx}\left[3x\int_{5x-1}^{x^3}\sin t dt+2x\int_{5x-1}^{x^3}\ln t dt\right]$$ You now have two simpler integrals.

The second option involves the second theorem of calculus and the chain rule, and it was answered here.

Calculate integrals: a) $\frac{d}{dx}\int_{5x-1}^{x^3}x\cdot(3\sin t+2\ln t)dt$ and b) $\frac{d}{dx}\ln(\int_{2}^{\cos x}(2t+\sin t)dt)$

2 Answers2