Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sqrt{\sin x} \cos ^{3} x d x\)

Short answer

\( \frac{2}{3} (\sin x)^{3/2} - \frac{2}{7} (\sin x)^{7/2} + C \).

Step-by-step solution

Identify the substitution

First, identify a useful substitution for solving the integral \( \int \sqrt{\sin x} \cos^3 x \, dx \). Here, it is helpful to use the substitution \( u = \sin x \). Then, \( du = \cos x \, dx \).

Rewrite the integral

Using the substitution from Step 1, rewrite the integral in terms of \( u \). So, \( \sqrt{\sin x} = \sqrt{u} \), and \( \cos^3 x \, dx = \cos^2 x \cdot \cos x \, dx \). Since \( \cos^2 x = 1 - \sin^2 x = 1 - u^2 \), the integral becomes \( \int \sqrt{u} (1-u^2) \, du \).

Simplify and separate the integral

Expand the expression under the integral: \( \int \sqrt{u} (1-u^2) \, du = \int (\sqrt{u} - u^{5/2}) \, du \). This separation allows us to integrate term by term.

Integrate each term

Now integrate each term separately: \( \int \sqrt{u} \, du = \int u^{1/2} \, du = \frac{2}{3}u^{3/2} + C \), and \( \int -u^{5/2} \, du = -\frac{2}{7}u^{7/2} + C \).

Combine and simplify

Combine the results from Step 4: \( \frac{2}{3} u^{3/2} - \frac{2}{7} u^{7/2} + C \).

Back-substitution

Substitute back \( u = \sin x \) to get the expression in terms of \( x \): \( \frac{2}{3} (\sin x)^{3/2} - \frac{2}{7} (\sin x)^{7/2} + C \).

Verification with CAS

Use a computer algebra system (CAS) to verify the result. Input the integral \( \int \sqrt{\sin x} \cos^3 x \, dx \) into the CAS and confirm that the solution matches: \( \frac{2}{3} (\sin x)^{3/2} - \frac{2}{7} (\sin x)^{7/2} + C \).

Key concepts

Trigonometric Substitution

Trigonometric substitution is a valuable technique in calculus when you encounter integrals involving square roots of trigonometric functions, such as \(\int \sqrt{\sin x} \cos^3 x \, dx\). The idea is to replace the trigonometric expression with a simpler variable, which makes the integral easier to solve. For this exercise, we replace \(\sin x\) with \(u\), leading to the substitution \(u = \sin x\). This also means \(du = \cos x \, dx\).Substitution transforms the original integral into a polynomial expression in terms of \(u\), \(\int \sqrt{u} (1 - u^2) \, du\). By simplifying the integral through substitution, previously formidable trigonometric integrals are reduced to polynomials, which are significantly easier to integrate term by term.Using substitutions effectively requires practice, as you must determine which substitution will simplify the problem best. Common cases involve expressions like \(\sqrt{1-\sin^2 x}\) or \(1-\cos^2 x\), where identities such as \(\cos^2 x = 1 - \sin^2 x\) are used to substitution advantage.

Definite and Indefinite Integrals

In calculus, integration is a key process that helps calculate accumulations and areas under curves. There are two main types: definite and indefinite integrals. The integral \(\int \sqrt{\sin x} \cos^3 x \, dx\) is an indefinite integral, as no limits of integration are specified. Indefinite integrals lead to a family of functions that differ by a constant, represented by \(C\).An indefinite integral focuses on finding the function that, when differentiated, yields the original integrand. For example, solving the indefinite integral of \(\int \sqrt{\sin x} \cos^3 x \, dx\) results in \(\frac{2}{3} (\sin x)^{3/2} - \frac{2}{7} (\sin x)^{7/2} + C\). This signifies all potential functions from which our original expression could be derived through differentiation.Conversely, definite integrals calculate the exact area under a curve within specific limits. Definite integrals have a numerical result, once limits are applied, unlike indefinite integrals that result in a formula plus a constant.

Computer Algebra Systems (CAS)

Computer Algebra Systems (CAS) are essential tools in verifying integral solutions and simplifying complex calculus procedures. These systems, such as Mathematica or Wolfram Alpha, allow learners to check their manual solutions against machine-generated results for accuracy.In the context of the integral \(\int \sqrt{\sin x} \cos^3 x \, dx\), a CAS can be used to enter the expression directly. It will return the simplified integration result, which can be compared to the manually solved solution, \(\frac{2}{3} (\sin x)^{3/2} - \frac{2}{7} (\sin x)^{7/2} + C\). This step helps not only in verifying answers but also in understanding interim calculations by showing how a system processes and breaks down the math.CAS are especially handy in educational contexts, offering error checking, step-by-step breakdowns, and a vast library of mathematical knowledge. They serve as a complementary learning aid, strengthening understanding through technology.