Question

Think About It Explain why the two integrals are equal. \(\int_{1}^{e} \sqrt{1+\frac{1}{x^{2}}} d x=\int_{0}^{1} \sqrt{1+e^{2 x}} d x\) Use the integration capabilities of a graphing utility to verify that the integrals are equal.

Short answer

The two integrals are equal because, on applying the substitution \(u = e^{x}\), the integrand of the second integral becomes exactly the same as the first one, over the same interval [1, e]. Thus the two integrals are equal.

Step-by-step solution

Recognise the integral structure.

Notice that the second integral has a slightly more complex structure due to the exponential function. However, both functions share a common denominator under the square root. This similarity might hint at a transformation of variables we can use to demonstrate the equality.

Decide on the set up for variable substitution.

In order to make the right hand side (RHS) integral look like the left hand side (LHS), let's substitute \(u = e^{x}\) in the RHS integral, which entails \(x = \ln u\) and \(dx = 1/u \, du\). Note that as \(x\) ranges from \(0\) to \(1\), \(u\) ranges from \(1\) to \(e\).

Carry out the variable substitution.

Substitute the new variables into the RHS integral: \[\int_{1}^{e} \sqrt{1 + u^{2}} \cdot \frac{1}{u} \, du\] Notice that after simplifying, this is exactly the same as the LHS integral.