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.