How do you integrate #int x^3/sqrt(x^2+10)dx# using trigonometric substitution?
Substitute:
So:
Use now the trigonometric identity:
and:
Substitute now:
to have:
and using the linearity of the integral:
undoing the substitution:
I would note however that even if the question required integration by trigonometric substitution, it would be easier to integrate differently:
Substitute:
so:
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To integrate ( \int \frac{x^3}{\sqrt{x^2+10}} , dx ) using trigonometric substitution, let ( x = \sqrt{10} \tan(\theta) ). Then, ( dx = \sqrt{10} \sec^2(\theta) , d\theta ). Substituting these into the integral gives:
[ \int \frac{x^3}{\sqrt{x^2+10}} , dx = \int \frac{(\sqrt{10} \tan(\theta))^3}{\sqrt{(\sqrt{10} \tan(\theta))^2+10}} (\sqrt{10} \sec^2(\theta)) , d\theta ]
[ = \int \frac{10\sqrt{10}\tan^3(\theta)}{\sqrt{10\tan^2(\theta)+10}} \cdot \sqrt{10}\sec^2(\theta) , d\theta ]
[ = \int \frac{10\sqrt{10}\tan^3(\theta)}{\sqrt{10(\tan^2(\theta)+1)}} \cdot \sqrt{10}\sec^2(\theta) , d\theta ]
[ = \int \frac{10\sqrt{10}\tan^3(\theta)}{\sqrt{10}\sec(\theta)} \cdot \sqrt{10}\sec^2(\theta) , d\theta ]
[ = \int 10\tan^3(\theta) \cdot \sec(\theta) , d\theta ]
Now, we can use the trigonometric identity ( \tan^2(\theta) = \sec^2(\theta) - 1 ) to simplify the integral:
[ = \int 10(\sec^2(\theta) - 1)\tan(\theta) \cdot \sec(\theta) , d\theta ]
[ = \int 10(\sec^3(\theta) - \sec(\theta)) , d\theta ]
[ = 10\int \sec^3(\theta) , d\theta - 10\int \sec(\theta) , d\theta ]
The integral ( \int \sec^3(\theta) , d\theta ) can be evaluated using the reduction formula for integrals of powers of secant. The integral ( \int \sec(\theta) , d\theta ) is a standard integral.
After finding these integrals and substituting back ( x = \sqrt{10} \tan(\theta) ), you will have the result in terms of ( x ) and ( \theta ).
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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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