Check the solution to "Evaluate and simplify the integral. ∫(x+2)x+1dx"

jernplate8

Answered question

2021-09-24

Evaluate and simplify the integral.

\int (x + 2) \sqrt{x + 1} d x

Arham Warner

Skilled2021-09-25Added 102 answers

Step 1
Consider the integrals:

\int (x + 2) \sqrt{x + 1} d x

...(1)
Let,

\sqrt{x + 1} = u

Squaring both sides,

x + 1 = u^{2} \Rightarrow x = u^{2} - 1

dx=2udu
Step 2
Substitute all value in equation (1) then,

\int (x + 2) \sqrt{x + 1} d x = \int (u^{2} - 1 + 2) u \cdot 2 u d u

= \int (u^{2} + 1) \cdot 2 u^{2} d u

= \int (2 u^{4} + 2 u^{2}) d u

= \frac{2 u^{5}}{5} + \frac{2 u^{3}}{3} + C

Back substitute

u = \sqrt{x + 1}

then,

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

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

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