∫(4x+2)√(x^2+x+1)dx Put, x^2 + x + 1 = t (2x + 1) dx = dt (4x + 2) dx = 2 dt Now, ∫ (4x + 2)√(x^2 + x + 1) dx = ∫ 2√(t) #160; dt ...

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Integration by Substitution

Solve ∫4x+2...

Question

Solve $\int (4 x + 2) \sqrt{x^{2} + x + 1} d x$

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Solution

$\int (4 x + 2) \sqrt{x^{2} + x + 1} d x$

Put, $x^{2} + x + 1 = t$

$⟹ (2 x + 1) d x = d t$

$⟹ (4 x + 2) d x = 2 d t$

Now, $\int (4 x + 2) \sqrt{x^{2} + x + 1} d x = \int 2 \sqrt{t} d t$

$= 2 \frac{(t)^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + c$

$= \frac{4}{3} t \sqrt{t} + c$

$= \frac{4}{3} (x^{2} + x + 1)^{\frac{3}{2}} + c$ (Ans)

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