Dive into the world of calculus as we solve the intriguing integral of ∫e⁽√ˣ⁾ dx. We’ll use powerful techniques like substitution and integration by parts to simplify and solve this complex integral. This journey illuminates the beauty of calculus, demonstrating its ability to tackle even challenging problems like ∫e⁽√ˣ⁾ dx. Perfect for students and math enthusiasts alike.

integral of e^sqrt(x) done step by step with substitution and integration by parts.
Etothesquarerootofxpart2

—-

————-

  1. Recognize the Integral: We start with the integral that we want to solve, which is \( \int e^{\sqrt{x}} dx \). This is not a standard form of integral that we can solve directly, so we need to use some techniques to simplify it.
  2. Choose a Substitution: We notice that the exponent of \(e\) is \(\sqrt{x}\), which suggests that a substitution involving \(\sqrt{x}\) might simplify the integral. So, we let \(u = \sqrt{x}\). This implies that \(x = u^2\).
  3. Find dx: To change the variable of integration from \(x\) to \(u\), we need to express \(dx\) in terms of \(du\). Differentiating \(x = u^2\) with respect to \(u\) gives \(dx = 2u du\).
  4. Transform the Integral: We substitute \(u\) for \(\sqrt{x}\) and \(dx = 2u du\) into the integral. This transforms the integral into \(\int e^u 2u du\). This new integral is easier to solve, as it is now in the form of an integral of \(u e^u\), which we can solve using integration by parts.
  5. Integration by Parts: The method of integration by parts is given by the formula \(\int u dv = uv – \int v du\). We choose \(u = 2u\) and \(dv = e^u du\). Then \(du = 2 du\) and \(v = e^u\). Applying the formula, we get \(2u e^u – \int 2 e^u du\).
  6. Solve the Simple Integral: The integral \(\int 2 e^u du\) is a standard integral, which is \(2e^u\). So, our original integral becomes \(2u e^u – 2e^u\).
  7. Back Substitution: Now we substitute \(u = \sqrt{x}\) back into the integral, yielding \(2\sqrt{x} e^{\sqrt{x}} – 2e^{\sqrt{x}}\), which is the indefinite integral of \(e^{\sqrt{x}}\).
  8. Add the Constant of Integration: The most general form of the integral includes a constant of integration, so the final answer is \(2\sqrt{x} e^{\sqrt{x}} – 2e^{\sqrt{x}} + C\), where \(C\) is the constant of integration. This represents the family of all antiderivatives of \(e^{\sqrt{x}}\).

—–

Plot of \(e^{\sqrt{x}}\)

Plot of e^sqrt(x)

Plot of the Integral of \(e^{\sqrt{x}}\)

Plot of Integral of e^sqrt(x)