Integral Of Sqrt 1 4x 2

Definite Integral of √(1 + 4x²)

The definite integral ∫√(1 + 4x²) dx presents a classic challenge in calculus, requiring a specific integration technique to solve. This comprehensive guide will walk you through the process, explaining the steps involved, the underlying concepts, and providing practical examples. We'll explore different approaches to tackling this integral, focusing on the trigonometric substitution method, a powerful tool for handling integrals involving square roots of quadratic expressions.

Understanding the Problem

Before diving into the solution, let's understand the integral itself: ∫√(1 + 4x²) dx. This integral represents the area under the curve of the function f(x) = √(1 + 4x²) from a lower limit to an upper limit (which will be specified later). The presence of the square root and the quadratic term (4x²) indicates that simple integration techniques won't suffice. We need a more sophisticated approach.

The Trigonometric Substitution Method

The most effective method for solving this type of integral is trigonometric substitution. This technique involves substituting trigonometric functions for the variable 'x' to simplify the expression under the square root. The specific substitution depends on the form of the quadratic expression. In this case, the expression resembles the Pythagorean identity: 1 + tan²θ = sec²θ. Therefore, we'll use the following substitution:

• 2x = tanθ

This implies:

• x = (1/2)tanθ
• dx = (1/2)sec²θ dθ

Now, let's substitute these expressions into the original integral:

∫√(1 + 4x²) dx = ∫√(1 + tan²θ) * (1/2)sec²θ dθ

Using the Pythagorean identity, we simplify the expression under the square root:

√(1 + tan²θ) = √(sec²θ) = secθ

The integral now becomes:

(1/2)∫secθ * sec²θ dθ = (1/2)∫sec³θ dθ

Solving the Integral of sec³θ

The integral of sec³θ is a recurring integral that requires a clever technique, often involving integration by parts. Let's break down the process:

Integration by Parts: The integration by parts formula is: ∫u dv = uv - ∫v du

Let's choose:

• u = secθ => du = secθ tanθ dθ
• dv = sec²θ dθ => v = tanθ

Applying the integration by parts formula:

(1/2)∫sec³θ dθ = (1/2)[secθ tanθ - ∫tanθ * secθ tanθ dθ]

Simplify the integral:

(1/2)[secθ tanθ - ∫secθ tan²θ dθ]

We can use the Pythagorean identity (tan²θ = sec²θ - 1) to further simplify:

(1/2)[secθ tanθ - ∫secθ (sec²θ - 1) dθ]

(1/2)[secθ tanθ - ∫sec³θ dθ + ∫secθ dθ]

Notice that the integral ∫sec³θ appears on both sides of the equation. Let's solve for this integral:

(1/2)∫sec³θ dθ = (1/2)secθ tanθ - (1/2)∫sec³θ dθ + (1/2)∫secθ dθ

Let I represent (1/2)∫sec³θ dθ:

I = (1/2)secθ tanθ - I + (1/2)∫secθ dθ

2I = (1/2)secθ tanθ + (1/2)∫secθ dθ

Remember that ∫secθ dθ = ln|secθ + tanθ| + C (where C is the constant of integration). Therefore:

2I = (1/2)secθ tanθ + (1/2)ln|secθ + tanθ| + C

I = (1/4)secθ tanθ + (1/4)ln|secθ + tanθ| + C

Substituting Back to 'x'

Now, we have the solution in terms of θ. We need to substitute back to the original variable 'x'. Recall our initial substitution: 2x = tanθ. This implies:

• tanθ = 2x
• secθ = √(1 + tan²θ) = √(1 + 4x²)

Substitute these back into our solution for I:

I = (1/4)√(1 + 4x²) * (2x) + (1/4)ln|√(1 + 4x²) + 2x| + C

Therefore, the final solution to the definite integral is:

∫√(1 + 4x²) dx = (1/2)x√(1 + 4x²) + (1/4)ln|2x + √(1 + 4x²)| + C

Definite Integrals: Adding Limits

The above solution is for the indefinite integral. To evaluate a definite integral, you need to specify the upper and lower limits of integration. For example, let's calculate the definite integral from 0 to 1:

∫₀¹ √(1 + 4x²) dx

We substitute the limits into our solution:

[(1/2)(1)√(1 + 4(1)²) + (1/4)ln|2(1) + √(1 + 4(1)²)|] - [(1/2)(0)√(1 + 4(0)²) + (1/4)ln|2(0) + √(1 + 4(0)²)|]

This simplifies to:

(1/2)√5 + (1/4)ln(2 + √5)

This provides a numerical approximation for the definite integral within those specific limits.

Applications and Further Exploration

This integral, and its solution technique, has numerous applications in various fields, including:

• Physics: Calculating arc length, work done, and other physical quantities.
• Engineering: Solving problems related to curves and surfaces.
• Computer Graphics: Generating curves and surfaces for modeling and animation.

Further exploration of this topic could include:

• Different trigonometric substitutions: Exploring other substitutions depending on the form of the quadratic.
• Hyperbolic substitutions: An alternative approach for certain types of integrals.
• Numerical integration methods: Using numerical techniques when analytical solutions are difficult to obtain.

This detailed explanation provides a comprehensive understanding of how to solve the integral of √(1 + 4x²), highlighting the power of trigonometric substitution and offering insights into its broader applications. Remember, practice is key to mastering these techniques. Work through various examples and explore different approaches to build your understanding and problem-solving skills.