Let $x > 0$. Prove that the value of $$\int^{x}_{0} \frac{1}{1+t^2}dt + \int^{\frac{1}{x}}_{0} \frac{1}{1+t^2}dt$$ does not depend on $x$.
I really have no idea how to start this problem. How do I show something doesnt "depend" on something?
Let $x > 0$. Prove that the value of $\int^{x}_{0} \frac{1}{1+t^2}dt + \int^{\frac{1}{x}}_{0} \frac{1}{1+t^2}dt$ does not depend on $x$.
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calculus
integration
definite-integrals
indefinite-integrals
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2Take the derivative with respect to $x$ would be a start. – 2017-02-09
1 Answers
2
In the second integral, let $t=1/u$ to get
$$\int_0^{1/x}\frac1{1+t^2}\ dt=\int_x^\infty\frac1{1+(1/u)^2}\frac{du}{u^2}=\int_x^\infty\frac1{1+u^2}\ du$$
Add it to the first integral to get
$$I=\int_0^\infty\frac1{1+t^2}\ dt=\frac\pi2$$
which does not depend on $x$.