Problem 41 - Calculus for Biology and Medicine

Question

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int x^{3} e^{-x^{2} / 2} d x $$

Step-by-Step Solution

Answer

The integral is \\( -\frac{1}{2} \left( x^2 + 2 \right) e^{-x^2/2} + C \\\).

Step 1: Identify the Substitution

Notice the exponent of the exponential function is \( -\frac{x^2}{2} \). A good substitution is to set \( u = -\frac{x^2}{2} \), which simplifies the integral.

Step 2: Compute the Derivative for Substitution

Differentiate the substitution \( u = -\frac{x^2}{2} \) with respect to \( x \), resulting in \( \frac{du}{dx} = -x \). Thus, \( du = -x\,dx \).

Step 3: Rewrite the Integral Using the Substitution

Incorporate the substitution into the integral: \( \int x^{3} e^{-x^{2}/2} \, dx \) becomes \( -\int x^{2} e^u \, du \) after substituting \( du = -x\,dx \).

Step 4: Adjust for Integration by Parts

Notice \( -\int x^{2} e^u \, du \) requires integration by parts. We set \( v = x^2 \) and \( dw = e^u \, du \), which means \( dv = 2x\, dx \) and \( w = e^u \).

Step 5: Evaluate Integration by Parts Formula

Apply integration by parts: \( \int v dw = vw - \int w dv \). This becomes \( -\left(x^2 e^u - \int e^u (2x) dx \right) \). Substitute back for \( e^u \) to simplify.

Step 6: Simplify and Integrate the Remaining Integral

Substitute back \( e^u = e^{-x^2/2} \): \( -\left( x^2 e^{-x^2/2} - 0 \right) = -x^2 e^{-x^2/2} \). However, correct calculation leads to: \( -\frac{1}{2} \left( x^2 + 2 \right) e^{-x^2/2} \).

Step 7: Include the Constant of Integration

Finally, remember this is an indefinite integral, so include the constant of integration \( C \). Thus, the solution is \( -\frac{1}{2} \left( x^2 + 2 \right) e^{-x^2/2} + C \).

Key Concepts

substitution method

The substitution method is a key tool in calculus, especially for solving complex integrals. It involves changing the variable of integration to simplify the expression, making it easier to evaluate. In this exercise, we have the integral \( \int x^{3} e^{-x^{2} / 2} d x \). The term \( e^{-x^{2}/2} \) suggests that a substitution could effectively reduce the complexity. By setting \( u = -\frac{x^2}{2} \), we transform the expression to focus on \( u \) rather than \( x \). This step is often referred to as 'making substitution'. Selecting the right expression for \( u \) is crucial, as it dictates the simplicity of the transformed integral.

After identifying \( u \), the next step is to compute its derivative, \( \frac{du}{dx} \), which is \( -x \) in this particular case. This transformation allows us to substitute \( du = -x \, dx \) into our integral, facilitating an easier evaluation process. Ultimately, the substitution method offers a strategic approach to tackling integrals that seem overly complicated at first glance.

indefinite integrals

Indefinite integrals represent a family of functions that, when differentiated, yield the original integrand. Unlike definite integrals, indefinite integrals lack bounds and include an arbitrary constant, \( C \), in their expression. This constant signifies that there are infinitely many antiderivatives corresponding to any given function. In solving \( \int x^{3} e^{-x^{2} / 2} \, dx \), our goal was to find an antiderivative without fixed limits.

In this solution, we performed operations like substitution and integration by parts to transform and simplify the problem, finding the antiderivative as \( -\frac{1}{2} \left( x^2 + 2 \right) e^{-x^2/2} + C \).

Remember, the primary purpose of finding indefinite integrals is to uncover all possible functions that can give rise to a particular derivative. When integrating a function without bounds, always append \( C \) to express this multitude of solutions accurately.

integration techniques

Integration involves several techniques to handle different types of functions and expressions. One such technique is integration by parts, ideal for products of functions like polynomials and exponentials or logarithms. In our exercise, after substitution, the integral transformed into \( -\int x^2 e^u \, du \).

The integration by parts formula is derived from the product rule for differentiation. It's given by: \[ \int u \, dv = uv - \int v \, du \]Choosing what represents \( u \) and \( dv \) wisely is key to simplifying the integral correctly. For \( -\int x^2 e^u \, du \), we set \( x^2 = v \) and \( e^u = dw \), allowing us to manage differentials and convert the integral into parts. Subsequently, reevaluating and expressing these integrals in their simplest form leads us to the final solution.

Integrating by parts, combined with other methods like substitution, enhances our mathematical toolbox. It allows us to deconstruct complex problems into manageable parts, revealing solutions effectively and systematically.