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How do you integrate by substitution #int tsqrt(t^2+2)dt#?

Answer 1

Cameron Alexander

The answer is #=(t^2+2)^(3/2)/3+C#

Let #u^2=t^2+2#

Differentiating both sides

#2udu=2tdt#, #=>#, #tdt=udu#

Therefore,

#inttsqrt(t^2+2)dt=intu*udu#

#intu^2du=u^3/3#

#=(t^2+2)^(3/2)/3+C#

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Answer 2

Luke Alvarez

To integrate ( \int t \sqrt{t^2 + 2} , dt ) by substitution, let ( u = t^2 + 2 ). Then ( du = 2t , dt ). Solving for ( dt ), we have ( \frac{du}{2t} = dt ). Substitute ( u ) and ( du ) into the integral and simplify. After integration, the result is ( \frac{1}{3}(t^2 + 2)^{\frac{3}{2}} + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.