Question

evaluate the indefinite integral.\\(\int x^4 \sqrt{14 + x^5} dx\\)\\(\square\\) + c

Explanation:

Step1: Choose substitution variable

Let $u = 14 + x^5$

Step2: Compute derivative of u

$\frac{du}{dx} = 5x^4 \implies du = 5x^4 dx \implies x^4 dx = \frac{1}{5}du$

Step3: Rewrite integral in terms of u

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

Step4: Integrate using power rule

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

Step5: Substitute back for x

Replace $u$ with $14 + x^5$

Answer:

$\frac{2}{15}(14 + x^5)^{\frac{3}{2}}$