Chanceling two definite integrals, that are same, but with different limits

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I have some stupid question. I am doing some assingment, and I have reached so far: $$ \int_{T/4}^0 \frac{4t}{T}~dt-\int_{T/4}^{T/2} \frac{4t}{T}~dt\\ $$ I see that this two integrals are the same. Is it possible that I just chancel them out.

Thanks

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2 Answers

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$$ \int_{T/4}^0 - \int_{T/4}^{T/2} \quad = \quad \int_{T/4}^0 + \int^{T/4}_{T/2} \quad = \quad \int_{T/2}^0 $$

In general, $$ \int_a^b+\int_b^c=\int_a^c. $$

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$$\begin{align*}\int\limits_{T/4}^0\frac{4t}Tdt&=\left.\frac2Tt^2\right|_{T/4}^0=-\frac2T\frac{T^2}{16}=-\frac T8\\\int\limits_{T/2}^{T/4}\frac{4t}Tdt&=\left.\frac2Tt^2\right|_{T/2}^{T/4}=\frac2T\left(\frac{T^2}{16}-\frac{T^2}4\right)=-\frac{3T}8\end{align*}$$

...so how are these two integrals the same?

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