How do you integrate #(sqrt(12+4x^2)dx#?

Answer 1
The answer is: #xsqrt(x^2+3)+3arc sinh(x/sqrt3)+c#.

The integral can be written:

#intsqrt(4(3+x^2))dx=2intsqrt(3+x^2)dx=(1)#

Since:

#x=sqrt3sinhtrArrdx=sqrt3coshtdt#, than:
#(1)=2intsqrt(3+3sinh^2t)*sqrt3coshtdt=#
#=2intsqrt3sqrt(1+sinh^2t)sqrt3coshtdt=6intcosht*coshtdt=#
#=6intcosh^2tdt=(2)#.

Now remembering that:

#cosh(alpha/2)=sqrt((coshalpha+1)/2)#,
#(2)=6int(cosh2t+1)/2dt=6/2(1/2int2cosh2t+intdt)=#
#=3(1/2sinh2t+t)+c=(3)#

And now, remembering that:

#sinh2alpha=2sinhalphacoshalpha# and #coshalpha=sqrt(sinh^2alpha+1)#,

than:

#(3)=3(1/2 2sinhtcosht+t)=#
#=3(x/sqrt3sqrt(x^2/3+1)+arc sinh(x/sqrt3))+c=#
#=3(x/sqrt3sqrt(x^2+3)/sqrt3+arc sinh(x/sqrt3))+c=#
#=xsqrt(x^2+3)+3arc sinh(x/sqrt3)+c#.
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Answer 2

To integrate √(12 + 4x^2) dx, you can use the trigonometric substitution method. Let x = √(3/2) * tan(θ), then dx = √(3/2) * sec^2(θ) dθ. After substitution and simplification, the integral becomes:

∫ √(12 + 4x^2) dx = ∫ √(12 + 4(√(3/2) * tan(θ))^2) * √(3/2) * sec^2(θ) dθ.

Simplify the expression inside the square root:

= ∫ √(12(1 + tan^2(θ))) * √(3/2) * sec^2(θ) dθ = ∫ √(12sec^2(θ)) * √(3/2) * sec^2(θ) dθ = ∫ √(12) * sec(θ) * √(3/2) * sec^2(θ) dθ = ∫ 2√(3) * sec^3(θ) dθ.

Now, use the integral of sec^3(θ) which is a known integral, leading to:

= 2√(3) * (1/2) * (sec(θ) * tan(θ) + ln|sec(θ) + tan(θ)|) + C = √(3) * (sec(θ) * tan(θ) + ln|sec(θ) + tan(θ)|) + C.

Finally, substitute back for θ using the original substitution x = √(3/2) * tan(θ), which implies tan(θ) = x/√(3/2) and sec(θ) = √(2/3) / √(1 - x^2/2), resulting in:

= √(3) * (√(2/3) / √(1 - x^2/2) * (x/√(3/2)) + ln|√(2/3) / √(1 - x^2/2) + (x/√(3/2))|) + C = (x√(2/3) / √(1 - x^2/2) + ln|√(2/3) / √(1 - x^2/2) + (x/√(3/2))|)√(3) + C.

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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.

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.

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.

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.

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