Evaluate the indefinite integral.
\int e^x \sqrt{14 + e^x} dx
Step 1
We must decide what to choose for u.
If u = f(x), then du = f'(x) dx, and so it is helpful to look for some expression in \int e^x \sqrt{14 + e^x} dx for which the derivative is also present.
We see that 14 + e^x is part of this integral, and the derivative of 14 + e^x is e^x which is also present.
Step 2
If we choose u = 14 + e^x, then du = e^x dx.
If u = 14 + e^x is substituted into \int e^x \sqrt{14 + e^x} dx, then we have
\int e^x \sqrt{14 + e^x} dx = \int \sqrt{u} dx = \int \sqrt{u} (e^x dx)
We must also convert e^x dx into an expression involving u, but we already know that e^x dx = \frac{2}{3} (14 + e^x)^{\frac{3}{2}} + c
