# How do you use substitution to integrate # 1/sqrt(x^2 -4)#?

ln I

= ln lsecu +tan u I +C

By signing up, you agree to our Terms of Service and Privacy Policy

To integrate ( \frac{1}{\sqrt{x^2 -4}} ) using substitution, we can use the trigonometric substitution method. Let ( x = 2\sec(\theta) ), then ( dx = 2\sec(\theta)\tan(\theta) , d\theta ).

Substituting ( x = 2\sec(\theta) ) into ( \sqrt{x^2 -4} ), we get ( \sqrt{(2\sec(\theta))^2 - 4} = \sqrt{4\tan^2(\theta)} = 2\tan(\theta) ).

Thus, ( \frac{1}{\sqrt{x^2 -4}} = \frac{1}{2\tan(\theta)} ).

Now, substitute ( dx = 2\sec(\theta)\tan(\theta) , d\theta ) and ( \frac{1}{\sqrt{x^2 -4}} = \frac{1}{2\tan(\theta)} ) into the integral:

( \int \frac{1}{\sqrt{x^2 -4}} , dx = \int \frac{1}{2\tan(\theta)} \cdot 2\sec(\theta)\tan(\theta) , d\theta ).

This simplifies to:

( \int d\theta ).

Now, integrating ( \int d\theta ) yields ( \theta + C ).

Finally, substitute back ( x = 2\sec(\theta) ) into ( \theta + C ) to get the final answer in terms of ( x ).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- How do you use Integration by Substitution to find #inttan(x)*sec^3(x)dx#?
- How do you integrate #int (3x^3+2x^2-7x-6)/(x^2-4) dx# using partial fractions?
- What is the antiderivative of #ln(1-x)/x#?
- How do you evaluate the integral #int dx/(x^3+x)#?
- How do you integrate #int (2x+1)/((x+4)(x-1)(x+7)) # using partial fractions?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7