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Gnom [1K]
2 years ago
10

10. Find

%5Csin%20t%7D_1%5Csqrt%7B1%2Bu%5E%7B4%7D%7Ddu%5Cright%29dt%5Ctext%7B.%7D" id="TexFormula1" title="\frac{d^{2}}{d x^{2}}\int ^x_0\left(\int ^{\sin t}_1\sqrt{1+u^{4}}du\right)dt\text{.}" alt="\frac{d^{2}}{d x^{2}}\int ^x_0\left(\int ^{\sin t}_1\sqrt{1+u^{4}}du\right)dt\text{.}" align="absmiddle" class="latex-formula">
Mathematics
1 answer:
RideAnS [48]2 years ago
3 0

Let g(t) denote the inner integral. By the fundamental theorem of calculus, the first derivative is

\displaystyle \frac{d}{dx} \int_0^x g(t) \, dt = g(x)

Then using the FTC again, differentiating g gives

\displaystyle \frac{dg}{dx} = \frac{d}{dx} \int_1^{\sin(x)} \sqrt{1+u^4} \, du = \boxed{\cos(x) \sqrt{1+\sin^4(x)}}

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