Question

$\bf \LARGE{Hola\ Guys!}$

Solve $\displaystyle \bf \int\ e^{\sqrt{x}}\ dx$

$\rm \Large{Thank\ you!}$

Answer 1

$\displaystyle \sf \int e^{\sqrt{x}}\ dx$

Let √x = t

⇒ x = t²

⇒ dx = 2t dt

$\longrightarrow \displaystyle \sf \int e^t\ (2t\ dt)$

$\longrightarrow\ \displaystyle \sf 2 \int e^t\ t\ dt$

Apply By Parts , (u=t & dv=eᵗdt)

⇒ du = dt & v = eᵗ

∫ udv = uv - ∫ vdu

$\longrightarrow\ \displaystyle \sf 2 \left( te^t\ - \int e^t\ dt \right)$

$\longrightarrow\ \displaystyle \sf 2 \left( te^t\ - e^t \right)$

$\longrightarrow\ \displaystyle \sf 2e^t \left( t\ - 1 \right)$

$\longrightarrow\ \displaystyle \pink{\boxed{\underline{ \sf2e^{\sqrt{x}} \left( \sqrt{x}\ - 1 \right) + c}}}$

Answer 2

This is your answer thankyou