Question

evaluate the indefinite integral.\\(\int x^3 \sqrt{5 + x^4} dx\\)\\(\boxed{}\\) + c\
question help: \\(\boxed{}\\) video \\(\boxed{}\\) message instructor\
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Explanation:

Step1: Choose substitution variable

Let $u = 5 + x^4$

Step2: Compute derivative of u

$\frac{du}{dx} = 4x^3 \implies du = 4x^3dx \implies x^3dx = \frac{1}{4}du$

Step3: Rewrite integral in terms of u

$\int \sqrt{u} \cdot \frac{1}{4}du = \frac{1}{4} \int u^{\frac{1}{2}}du$

Step4: Integrate using power rule

$\frac{1}{4} \cdot \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{1}{4} \cdot \frac{2}{3}u^{\frac{3}{2}} = \frac{1}{6}u^{\frac{3}{2}}$

Step5: Substitute back u

$\frac{1}{6}(5 + x^4)^{\frac{3}{2}}$

Answer:

$\frac{1}{6}(5 + x^4)^{\frac{3}{2}}$