How to solve the integration question #int1/(4+9x^2)^(1/2)# using trigonometric substitution?
#int1/(4+9x^2)^(1/2)#
Substitute:
we have:
Use now the trigonometric identity:
and:
we do not simplify but substitute again:
which can be solved by partial fractions:
so:
and using the properties of logarithms:
Reversing the substitution we have:
So:
Rationalize the denominator of the argument:
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To solve the integral ∫(1 / √(4 + 9x^2)) dx using trigonometric substitution, follow these steps:
-
Recognize the form of the integral and choose the appropriate trigonometric substitution. In this case, since we have 4 + 9x^2, it suggests using x = (2/3)tan(θ).
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Compute dx in terms of dθ using the derivative of the substitution x = (2/3)tan(θ).
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Substitute x and dx in terms of θ into the integral.
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Simplify the integrand using trigonometric identities to express it solely in terms of θ.
-
Integrate the expression with respect to θ.
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Finally, revert back to the original variable x using the trigonometric identities and simplify the result.
By following these steps, you can solve the given integral using trigonometric substitution.
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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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