Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
The integral of \( \frac{e^{\sqrt{x}}}{\sqrt{x}} \) with respect to \( x \) is \( 2e^{\sqrt{x}} + C \).
Solution
Calculate the integral \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \).
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
\(\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx\)
- step1: Rewrite the expression:
\(\int \frac{e^{\left(x^{\frac{1}{2}}\right)}}{x^{\frac{1}{2}}} dx\)
- step2: Use the substitution \(dx=2x^{\frac{1}{2}} dt\) to transform the integral\(:\)
\(\int \frac{e^{\left(x^{\frac{1}{2}}\right)}}{x^{\frac{1}{2}}}\times 2x^{\frac{1}{2}} dt\)
- step3: Simplify:
\(\int 2e^{\left(x^{\frac{1}{2}}\right)} dt\)
- step4: Use the substitution \(t=x^{\frac{1}{2}}\) to transform the integral\(:\)
\(\int 2e^{t} dt\)
- step5: Use properties of integrals:
\(2\times \int e^{t} dt\)
- step6: Evaluate the integral:
\(2e^{t}\)
- step7: Substitute back:
\(2e^{\left(x^{\frac{1}{2}}\right)}\)
- step8: Simplify:
\(2e^{\sqrt{x}}\)
- step9: Add the constant of integral C:
\(2e^{\sqrt{x}} + C, C \in \mathbb{R}\)
The integral of \( \frac{e^{\sqrt{x}}}{\sqrt{x}} \) with respect to \( x \) is \( 2e^{\sqrt{x}} + C \), where \( C \) is an arbitrary constant.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To solve the integral \( \int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x \), we can use a substitution method. Let’s set \( u = \sqrt{x} \), hence \( x = u^2 \) and \( dx = 2u \, du \). This transforms the integral into: \[ \int \frac{e^{u}}{u} \cdot 2u \, du = 2 \int e^{u} \, du \] Now, integrating \( e^{u} \) gives us: \[ 2 e^{u} + C \] Finally, we substitute back \( u = \sqrt{x} \) to obtain: \[ 2 e^{\sqrt{x}} + C \] So, the solution to the original integral is \( 2 e^{\sqrt{x}} + C \).
