Problem 44 - Calculus for Biology and Medicine

Question

First make an appropriate substitution and then use integration by parts to evaluate the indefinite integrals. $$ \int \sin x \cos ^{3} x e^{1-\sin ^{2} x} d x $$

Step-by-Step Solution

Answer

Use the substitution \( u = \sin x \), and then proceed with integration by parts on the transformed function.

Step 1: Make a Substitution for Simplification

We start by observing the function inside the integral. Notice that \( e^{1 - \sin^2 x} \) can be simplified using the identity \( 1 - \sin^2 x = \cos^2 x \). Hence, let's make the substitution \( u = \sin x \), which implies \( du = \cos x \, dx \). This changes our integral into: \[ \int \sin x \cos^3 x \, e^{\cos^2 x} \, dx = \int u(1-u^2)^2 e^{(1-u^2)} \, du.\]

Step 2: Express in Terms of u

After substitution, the integral becomes: \[\int u(1-u^2)e^{(1-u^2)} (1-u^2) du \] Expanding \((1-u^2)^2 \) gives us \( (1 - 2u^2 + u^4) \), so the integral is now: \[\int u(1 - 2u^2 + u^4)e^{(1-u^2)} \, du \]Distributing \(u\) gives us: \[ \int (u - 2u^3 + u^5) e^{(1-u^2)} \, du. \]

Step 3: Apply Integration by Parts

Integration by parts formula is \( \int v \, du = uv - \int u \, dv \). Let's choose \( v = u - 2u^3 + u^5 \) and \( du = e^{(1-u^2)} \, du \). The derivative \( dv = (1 - 6u^2) du \), so integrate using this formula.

Key Concepts

Substitution Method

The substitution method is a powerful technique in calculus used to simplify integration. It involves changing the variable of integration to transform a difficult integral into a more manageable one. The key is to identify a part of the integrand that can be set equal to a new variable, which we call "u". This is similar to the idea of using a substitution in algebra to solve an equation.

In the original exercise, the expression inside the exponential, \( e^{1 - \sin^2 x} \), is tricky to integrate directly. By recognizing that \( 1 - \sin^2 x = \cos^2 x \) and letting \( u = \sin x \), we streamline the integral. When we differentiate \( u = \sin x \), we get \( du = \cos x \, dx \). This substitution not only simplifies the integral but also changes its boundaries and differentials.

To apply this method effectively:

• Identify a substitution that simplifies part of the integrand.
• Calculate the differential of the substituted variable.
• Rewrite the entire integral in terms of the new variable.

This transforms the original integral into a simpler form, making subsequent steps more straightforward.

Indefinite Integrals

Indefinite integrals represent the family of all antiderivatives of a given function. When you evaluate an indefinite integral, you find not just a single function, but a collection of functions differing by a constant. This is because when you differentiate a function, any constant term disappears, making it indistinct in the resulting derivative.

In our integral, we denote it by:\[\int f(x) \, dx = F(x) + C\]where \(F(x)\) is an antiderivative of \(f(x)\), and \(C\) is the arbitrary constant.

Understanding indefinite integrals is critical because when solving a problem using integration, we often need to obtain a general form of solutions. For the exercise provided, the indefinite integral \( \int \sin x \cos^3 x e^{1-\sin^2 x} \ dx \) must be computed using techniques like substitution and integration by parts, and the answer will involve a "+ C" to denote the family of solutions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the identity are defined. They are essential tools for simplifying expressions and solving integrals involving trigonometric functions.

In the context of our exercise, the identity \( 1 - \sin^2 x = \cos^2 x \) is crucial. This particular identity helps in transforming the exponential expression \( e^{1 - \sin^2 x} \) into \( e^{\cos^2 x} \), which is much easier to handle from an algebraic standpoint. Employing such identities allows us to rewrite the integral in a more convenient form for either direct integration or further substitutions.

Here's how trigonometric identities typically assist in integrals:

• Simplifying expressions by rewriting trigonometric functions.
• Reducing complex functions into standard forms.
• Facilitating substitution methods by replacing complicated expressions.

Ink exercises involve not only mechanical application of these techniques but also a deeper recognition of how trigonometric identities enable different paths of problem-solving.