Problem 60 - Calculus Single Variable

Question

Calculate the given integral. $$ \int 2 \sqrt{x^{2}+2 x+2} d x $$

Step-by-Step Solution

Answer

\( \arcsinh(x+1) + (x+1)\sqrt{x^2+2x+2} + C \)

Step 1: Simplify the Expression Inside the Square Root

First, observe that the expression inside the square root can be rewritten to complete the square. The original expression is \[ x^2 + 2x + 2 \]We complete the square:\[ x^2 + 2x + 2 = (x+1)^2 + 1 \]

Step 2: Use a Substitution to Simplify the Integral

Let us perform the substitution:\[ u = x + 1 \]Then, we have:\[ du = dx \]And the expression inside the square root becomes:\[ \sqrt{u^2 + 1} \] This transforms the integral to:\[ \int 2 \sqrt{u^2 + 1} \, du \]

Step 3: Use Hyperbolic Substitution

For the term \( \sqrt{u^2 + 1} \), we can use a hyperbolic substitution. Set:\[ u = \sinh(t) \] Then \[ \cosh(t) = \sqrt{\sinh^2(t) + 1} \].Therefore, the differential \( du = \cosh(t) \, dt \), and the integral becomes:\[ \int 2 \cosh^2(t) \, dt \]

Step 4: Simplify and Integrate

Recall the identity:\[ \cosh^2(t) = \frac{1 + \cosh(2t)}{2} \]Then our integral becomes:\[ \int 2 \left( \frac{1 + \cosh(2t)}{2} \right) \, dt = \int (1 + \cosh(2t)) \, dt \]This simplifies to:\[ \int 1 \, dt + \int \cosh(2t) \, dt \]Integrating each term, we obtain:\[ t + \frac{1}{2} \sinh(2t) + C \]

Step 5: Reverse the Substitution

Recall the hyperbolic substitution \( u = \sinh(t) \), so \( t = \arcsinh(u) \).Substitute back to find expressions in terms of \( u \):\[ \int 2 \sqrt{u^2+1} \, du = \arcsinh(u) + \frac{1}{2} \sinh(2\arcsinh(u)) + C \].Now use the trigonometric identity \( \sinh(2t) = 2 \sinh(t) \cosh(t) \).Substitute back \( u = x + 1 \) to finalize the answer:\[ \arcsinh(x+1) + \sinh(t)(x+1)\cosh(t) + C \]

Step 6: Simplify the final expression

Simplify the final expression to remove hyperbolic functions when possible. Recall that \( \cosh(\arcsinh(x)) = \sqrt{x^2+1} \) and \( \sinh(\arcsinh(x)) = x \).Thus, we have:\[ \int 2 \sqrt{u^2 + 1} \, du = \arcsinh(x+1) + (x+1) \sqrt{x^2 + 2x + 2} + C \]Thus, the solution to the integral problem, in terms of original variable \( x \), is:\[ \arcsinh(x+1) + (x+1)\sqrt{x^2+2x+2} + C \]

Step 7: Final Step: Write Down the Final Answer

After all substitution steps and simplifications, the integral of the given function is:\[ \int 2 \sqrt{x^2 + 2x + 2} \, dx = \arcsinh(x+1) + (x+1)\sqrt{x^2 + 2x + 2} + C \]

Key Concepts

Integration Techniques

Integration techniques are crucial tools in calculus that help solve integrals which would otherwise be impossible to solve by basic integration. Each technique is crafted to handle specific types of integral problems. In this context, the integral we're discussing involves a complex expression under a square root sign.

• Direct Integration: This technique works for simple functions where antiderivatives are straightforward. However, as the complexity of the function increases, more sophisticated methods become necessary.
• Substitution Technique: Here, we aim to transform a complicated integral into a simpler one by changing variables. The idea is to substitute a part of the integral with a new variable, often simplifying the integral's form.
• Particular Functions: Functions like trigonometric, logarithmic, or in our case, hyperbolic, often require their own specific methods for integration, taking advantage of their properties.

Using the right technique can help tremendously in solving an integral correctly and efficiently.

Substitution Method

The substitution method is like a puzzle solver for integration problems. Its goal is to make an integral easier to handle by replacing a part of the expression with a new variable. In the exercise above, we wanted to simplify the expression under the square root. So, we used the substitution method:

• Identify the Inner Function: The expression inside the square root, \(x^2 + 2x + 2\), can be tricky to integrate directly.
• Complete the Square: This step involves rewriting \(x^2 + 2x + 2\) as \((x+1)^2 + 1\). Completing the square helps in revealing a structure which can be easily transformed.
• Substitute: Set \(u = x + 1\), which transforms the expression and the variable of integration to \(\sqrt{u^2 + 1}\) and \(du = dx\) respectively.

This substitution gives us a clearer path in tackling the integral by simplifying it significantly.

Hyperbolic Functions

Hyperbolic functions like \(\sinh, \cosh\), and \(\tanh\) mirror trigonometric functions but are based on hyperbolas instead of circles. They prove useful in integrals where expressions resemble hyperbolic identities. In our integral solution:

• Hyperbolic Substitution: The integral was simplified using a hyperbolic identity. By setting \(u = \sinh(t)\), the term \(\sqrt{u^2 + 1}\) was transformed to \(\cosh(t)\), thanks to the identity \(\cosh^2(t) - \sinh^2(t) = 1\).
• Hyperbolic Identity Utilization: Recognizing and utilizing \(\cosh^2(t) = \frac{1 + \cosh(2t)}{2}\) enabled further simplification. This substitution aligns well with the reshaped integral format.

These properties allow us to break down the problem using known identities and relationships of hyperbolic functions for successful integration.

Completing the Square

Completing the square is a technique used to simplify an expression that includes a quadratic polynomial. It transforms the polynomial into a perfect square plus or minus a constant, making it easier to work with, especially when under a square root. In our problem:

• Step-by-Step Transformation: Taking the expression \(x^2 + 2x + 2\) and rewriting it as \((x+1)^2 + 1\), provides a more straightforward basis for further manipulation.
• Advantages: This allows the expression to become more manageable, particularly when it is part of an integral, potentially enabling other techniques such as substitutions and hyperbolic transformations.
• Connection to Other Techniques: Completing the square can be the precursor for the substitution method, easing the transition to a more workable form of the integral.

Mastering this technique can provide clarity in how complex integrals are simplified through strategic mathematical manipulation.