How do you integrate #int sqrt(x^2+1)# by trigonometric substitution?
See answer below:
By signing up, you agree to our Terms of Service and Privacy Policy
To integrate ∫√(x^2 + 1) dx using trigonometric substitution, let x = tan(θ).
Then, dx = sec^2(θ) dθ, and √(x^2 + 1) = √(tan^2(θ) + 1) = sec(θ).
So the integral becomes ∫sec^2(θ) * sec(θ) dθ.
This simplifies to ∫sec^3(θ) dθ.
Now, you can use the reduction formula for ∫sec^n(θ) dθ:
∫sec^n(θ) dθ = (1/(n-1)) * sec^(n-2)(θ) * tan(θ) + (n-2)/(n-1) * ∫sec^(n-2)(θ) dθ
For n = 3, this becomes:
∫sec^3(θ) dθ = (1/2) * sec(θ) * tan(θ) + (1/2) * ∫sec(θ) dθ
Now, integrate ∫sec(θ) dθ using the substitution u = tan(θ):
∫sec(θ) dθ = ∫(1/cos(θ)) dθ = ∫(1/(1 + u^2)) du
This integral can be solved using a standard integral formula or by partial fractions.
Finally, substitute back θ = tan^(-1)(x) into the result obtained above 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.
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 #int 1/sqrt(7-3x^2)# by trigonometric substitution?
- How do you integrate #int (6x^3)/sqrt(9+x^2) dx# using trigonometric substitution?
- How do you find the integral of #f(x)=sin^2x# using integration by parts?
- How do you find #int (x-1) / ((2x+1) (4x+4)) dx# using partial fractions?
- How do you integrate #int tcsctcott# by parts?

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