Integrate dx/(sqrt(x^2+16)).

I have no idea how to start and which method to use. Thinking some sort of trig substitution? But it doesn't look like it.

Step by step? Answer key says ln|x+ sqrt(x^2+16)|.

Found this youtube for you

He is doing dx/√(x^2+a^2)
so for yours a^2 = 16, a = 4

https://www.youtube.com/watch?v=MWfA85-Mb50

Have fun

To solve the integral ∫dx/(√(x^2+16)), you can use a trigonometric substitution. However, in this case, we can use a simpler approach by recognizing that the function inside the square root, x^2+16, is a perfect square.

Let's go through the solution step by step:

Step 1: Identify the function parts
The integral can be rewritten as:
∫dx/(√(x^2+16))

Step 2: Simplify the expression inside the square root
Since x^2+16 is a perfect square, we can simplify it to:
x^2+16 = (x^2+4^2)

Step 3: Apply a trigonometric substitution
Since x^2+4^2 resembles the form of a trigonometric identity (a^2+b^2), we can use the trigonometric substitution x = 4tanθ. To do this, we need to find dx in terms of dθ. Taking the derivative of both sides gives:

dx = 4sec^2θ dθ

Step 4: Rewrite the integral with the trigonometric substitution
Substituting x = 4tanθ and dx = 4sec^2θ dθ, the integral becomes:

∫(4sec^2θ dθ)/(√((4tanθ)^2+16))
= ∫4sec^2θ dθ/(√(16tan^2θ+16))
= ∫4sec^2θ dθ/(√(16(tan^2θ+1)))

Step 5: Simplify the expression inside the square root
Using the trigonometric identity tan^2θ+1 = sec^2θ, we can simplify the expression under the square root. The integral becomes:

∫4sec^2θ dθ/(√(16(sec^2θ)))
= ∫4sec^2θ dθ/(√(16sec^2θ))
= ∫4sec^2θ dθ/(4secθ)

Step 6: Cancel out common factors
Simplifying further, we can divide both the numerator and the denominator by 4:

∫sec^2θ dθ/(secθ)
= ∫secθ dθ

Step 7: Integral of secθ
The integral of secθ can be evaluated using logarithmic properties. The antiderivative of secθ is ln|secθ + tanθ|. Therefore, our integral becomes:

∫dθ = ln|secθ + tanθ| + C

Step 8: Substituting back the value of x
Finally, substitute the value of x back into the solution. Recall that x = 4tanθ:

ln|secθ + tanθ| + C
= ln|sec(arctan(x/4)) + tan(arctan(x/4))| + C
= ln|√(x^2+16)/4 + x/4| + C
= ln|x + √(x^2+16)|/4 + C

Therefore, the solution to the integral ∫dx/(√(x^2+16)) is ln|x + √(x^2+16)| + C.