Get the solution to "Evaluate the following integral. ∫x3x2+4"

Anish Buchanan

Answered question

2021-10-25

Evaluate the following integral.

\int x^{3} \sqrt{x^{2} + 4}

Adnaan Franks

Skilled2021-10-26Added 92 answers

Step 1
Consider the integral:

\int x^{3} \sqrt{x^{2} + 4}

...(1)
To solve the given integrals,
Let,

\sqrt{x^{2} + 4} = t

x^{2} + 4 = t^{2}

(Squaring both sides)

x^{2} = t^{2} - 4

2xdx=2tdt-0
xdx=tdt
Step 2
Substitute all value in equation (1) then,

\int x^{3} \sqrt{x^{2} + 4} d x = \int x^{2} \sqrt{x^{2} + 4} \cdot x d x

= \int (t^{2} - 4) \cdot t \cdot t d t

= \int (t^{2} - 4) \cdot t^{2} \cdot d t

= \int (t^{4} - 4 t^{2}) d t

Further solving it,

= [\frac{t^{4 + 1}}{4 + 1} - 4 (\frac{t^{2} + 1}{2 + 1})] + C

= \frac{t^{5}}{5} - \frac{4 t^{3}}{3} + C

= \frac{1}{5} {(\sqrt{x^{2} + 4})}^{5} - \frac{4}{3} {(\sqrt{x^{2} + 4})}^{3} + C

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