Question

solve the integral \\(\int e^{x}\sqrt{5e^{x} + 4}\\ dx\\) with the following options:

a.) \\(\frac{2}{15}(5e^{x}+4)^{\frac{3}{2}} + c\\)

b.) \\(\frac{3}{10}e^{x}(5e^{x}+4)^{\frac{3}{2}} + c\\)

c.) \\(\frac{2}{15}e^{x}(5e^{x}+4)^{\frac{3}{2}} + c\\)

d.) \\(\frac{3}{10}(5e^{x}+4)^{\frac{3}{2}} + c\\)

Explanation:

Step1: Choose substitution

Let $u = 5e^x + 4$.

Step2: Compute derivative of u

$\frac{du}{dx} = 5e^x \implies du = 5e^x dx \implies e^x dx = \frac{1}{5}du$

Step3: Rewrite integral in u

$\int \sqrt{u} \cdot \frac{1}{5}du = \frac{1}{5}\int u^{\frac{1}{2}}du$

Step4: Integrate using power rule

$\frac{1}{5} \cdot \frac{u^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{1}{5} \cdot \frac{2}{3}u^{\frac{3}{2}} + C = \frac{2}{15}u^{\frac{3}{2}} + C$

Step5: Substitute back u

$\frac{2}{15}(5e^x + 4)^{\frac{3}{2}} + C$

Answer:

a.) $\frac{2}{15}(5e^{x}+4)^{\frac{3}{2}}+c$