# How do you integrate #int (4x)/sqrt(x^2-49)dx# using trigonometric substitution?

Hopefully this helps!

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

To integrate ( \int \frac{4x}{\sqrt{x^2 - 49}} , dx ) using trigonometric substitution, we can let ( x = 7\sec(\theta) ). Then, ( dx = 7\sec(\theta)\tan(\theta) , d\theta ).

Substituting these expressions into the integral, we get:

[ \int \frac{4 \cdot 7\sec(\theta)\tan(\theta)}{\sqrt{(7\sec(\theta))^2 - 49}} , d\theta ]

[ = \int \frac{28\sec(\theta)\tan(\theta)}{\sqrt{49\sec^2(\theta) - 49}} , d\theta ]

[ = \int \frac{28\sec(\theta)\tan(\theta)}{\sqrt{49(\sec^2(\theta) - 1)}} , d\theta ]

[ = \int \frac{28\sec(\theta)\tan(\theta)}{\sqrt{49\tan^2(\theta)}} , d\theta ]

[ = \int \frac{28\sec(\theta)\tan(\theta)}{7\tan(\theta)} , d\theta ]

[ = 4\int d\theta ]

[ = 4\theta + C ]

Finally, substituting back ( \theta = \sec^{-1}(\frac{x}{7}) ), we have:

[ = 4\sec^{-1}\left(\frac{x}{7}\right) + C ]

So, ( \int \frac{4x}{\sqrt{x^2 - 49}} , dx = 4\sec^{-1}\left(\frac{x}{7}\right) + C ).

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 #x^3 cos(x^2) dx#?
- How do you integrate #int 1/sqrt(x^2+1)# by trigonometric substitution?
- How do you integrate #int lnx/x^7# by integration by parts method?
- How do you integrate #(y^2 + 1) / (y^3 - 1)# using partial fractions?
- How do you integrate #int x/((x^2+1)^2)# using partial fractions?

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