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sineoko [7]
2 years ago
7

Calculus help ASAP!!!

Mathematics
1 answer:
anastassius [24]2 years ago
7 0
Let c\in(0,\infty). Then the definite integral can be split up at t=c so that

\displaystyle F(x)=\int_{t=2x}^{t=5x}\frac{\mathrm dt}t
\displaystyle=\int_{t=2x}^{t=c}\frac{\mathrm dt}t+\int_{t=c}^{t=5x}\frac{\mathrm dt}t
\displaystyle=-\int_{t=c}^{t=2x}\frac{\mathrm dt}t+\int_{t=c}^{t=5x}\frac{\mathrm dt}t

Now take the derivative. By the fundamental theorem of calculus, you have

\displaystyle\frac{\mathrm dF}{\mathrm dx}=\frac{\mathrm d}{\mathrm dx}\left[\int_{t=c}^{t=5x}\frac{\mathrm dt}t-\int_{t=c}^{t=2x}\frac{\mathrm dt}t\right]
=\displaystyle\frac1{5x}\dfrac{\mathrm d}{\mathrm dx}[5x]-\frac1{2x}\dfrac{\mathrm d}{\mathrm dx}[2x]
=\dfrac5{5x}-\dfrac2{2x}
=\dfrac1x-\dfrac1x
=0

Then integrating with respect to x, we recover F(x) and find that

\displaystyle\int\dfrac{\mathrm dF}{\mathrm dx}\,\mathrm dx=\int0\,\mathrm dx
F(x)=C

where C is an arbitrary constant.
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