Indefinite Integrals Using U-Substitution

1. Use \sqrt{x} to get   $$\sqrt{x}$$

2. Use the substitution:

  $$u=\sqrt{x}$$

$$\\du=\frac{dx}{2\sqrt{x}}\\2\sqrt{x}du=dx\\2udu=dx$$

So:

$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=\int \frac{e^{u} \times 2udu}{u}$$

or:

$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx=2\int e^{u}du = 2e^u=2e^{\sqrt{x}}$$

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Ugh, now this seems way too simple. Thank you a ton!