# How do you integrate #intxsec^-1(x)dx#?

The answer is

Perform the integration by parts

Here,

Therefore,

Therefore,

And finally,

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

To integrate ( \int x \sec^{-1}(x) , dx ), we use integration by parts. Let ( u = \sec^{-1}(x) ) and ( dv = x , dx ). Then, ( du = \frac{1}{|x|\sqrt{x^2 - 1}} , dx ) and ( v = \frac{1}{2}x^2 ).

Applying integration by parts formula ( \int u , dv = uv - \int v , du ), we get:

( \int x \sec^{-1}(x) , dx = \frac{1}{2}x^2 \sec^{-1}(x) - \frac{1}{2} \int \frac{x^2}{|x|\sqrt{x^2 - 1}} , dx ).

The remaining integral can be simplified using trigonometric substitution. Let ( x = \sec(\theta) ), then ( dx = \sec(\theta) \tan(\theta) , d\theta ). This simplifies the integral to:

( \int \frac{\sec^2(\theta)}{\sec(\theta)|\tan(\theta)|} , \sec(\theta) \tan(\theta) , d\theta ).

Simplify this expression and integrate it. Finally, substitute back ( \theta ) with ( \sec^{-1}(x) ) to get the final result.

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 integrate by substitution #int root3(1-2x^2)(-4x)dx#?
- How do you integrate #int x arcsec x # using integration by parts?
- How do you integrate #int (4x+8)/(x^2+4x-25)# using substitution?
- How do you integrate #int x^3 t an x dx # using integration by parts?
- How do you integrate #int x^3 ln x^4 dx # using integration by parts?

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