Question

Integrating \(\int \cos ^{j} x \sin ^{k} x d x\) where \(k\) and \(j\) are Even Evaluate \(\int \sin ^{4} x d x\).

Short answer

\(\int \sin^4 x \, dx = \frac{3}{8}x - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C\).

Step-by-step solution

Use Trigonometric Identity

To solve the integral of \(\int \sin^4 x \, dx\), we first use the trigonometric identity for \(\sin^2 x\):\[\sin^2 x = \frac{1 - \cos(2x)}{2}.\]Substitute this identity into the integral for \(\sin^4 x\):\[\sin^4 x = (\sin^2 x)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2.\]This converts the integral into a more manageable form.

Simplify and Expand

Now, expand the expression obtained in Step 1:\[\left(\frac{1 - \cos(2x)}{2}\right)^2 = \left(\frac{1}{2}(1 - \cos(2x))\right)^2 = \frac{1}{4}(1 - 2\cos(2x) + \cos^2(2x)).\]After expansion, your integral \(\int \sin^4 x \, dx\) becomes:\[\int \frac{1}{4}(1 - 2\cos(2x) + \cos^2(2x)) \, dx.\]

Simplify Further Using Another Identity

Use the identity for \(\cos^2(2x)\):\[\cos^2(2x) = \frac{1 + \cos(4x)}{2}.\]Substitute this into the integral:\[\int \frac{1}{4}(1 - 2\cos(2x) + \frac{1 + \cos(4x)}{2}) \, dx.\]This becomes:\[\int \frac{1}{4} \left(1 - 2\cos(2x) + \frac{1}{2} + \frac{1}{2}\cos(4x)\right) \, dx.\]

Separate and Integrate

The integral can now be separated and simplified as:\[\int \frac{3}{8} - \frac{1}{2}\cos(2x) + \frac{1}{8}\cos(4x) \, dx.\]Integrate each term separately:1. \(\int \frac{3}{8} \, dx = \frac{3}{8}x\)2. \(\int -\frac{1}{2}\cos(2x) \, dx = -\frac{1}{4}\sin(2x)\)3. \(\int \frac{1}{8}\cos(4x) \, dx = \frac{1}{32}\sin(4x)\).

Combine Results and Add Constant of Integration

Combine the results from Step 4 and add the constant of integration \(C\):\[\int \sin^4 x \, dx = \frac{3}{8}x - \frac{1}{4}\sin(2x) + \frac{1}{32}\sin(4x) + C.\]This is the final integrated form of the expression.

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in simplifying and solving integrals involving trigonometric functions. These identities relate the trigonometric functions to each other, allowing us to express complex trigonometric expressions in simpler forms.

One important identity used here is the double-angle identity for sine:

• \[\sin^2 x = \frac{1 - \cos(2x)}{2}\]

This identity allows us to transform powers of sine or cosine into equations involving angles that are easier to integrate. By applying such identities, we can rewrite integrals in a way that makes them more approachable.

These identities help in breaking down higher powers. Therefore, knowing your trigonometric identities can make your life a lot easier when dealing with complex integrals.

Even Powers Integrals

Integrals involving even powers of trigonometric functions like sine and cosine benefit greatly from trigonometric identities. When the powers are even, we can often use identities to transform the expression into a simpler form that is more straightforward to integrate.

For instance, in the integral \(\int \sin^4 x \, dx\), the power of 4 is even. This means we can use trigonometric identities to split or simplify the quadrant power into a combination of products or sums of terms with smaller powers.

• We start by transforming \(\sin^4 x\) into \((\sin^2 x)^2\).
• Next, substitute using the identity for \(\sin^2 x\) to make the expression manageable.

Understanding and manipulating even powers effectively is crucial in simplifying the integration process.

Trigonometric Integration

Trigonometric integration is the process used to evaluate integrals containing trigonometric functions. When dealing with such integrals, recognizing applicable trigonometric identities can simplify calculations significantly.

The example integral \(\int \sin^4 x \, dx\) demonstrates a typical scenario where transforming powers of sine using trigonometric identities is essential to progress in the solution. After applying an identity:

• Expand and distribute terms to make it feasible for direct integration.

The properties of trigonometric functions and their integrals provide a systematic approach to solve what might initially seem complex. Proficiency in this area is developed by practice and familiarity with identities, enabling smooth tackling of similar problems.

Integration by Substitution

Integration by substitution is a technique often used when simplifying expressions before integration. It is similar to the "reverse chain rule" in differentiation and involves changing the variable in an integral.

In this integral, initial steps involve substitution with identities rather than variable replacement to simplify the integrals. However, it shows the concept of transforming complex integrals into simpler ones which is foundational to substitution.

• This integration ultimately leads to splitting the integral and addressing each part separately.
• Once simplified, each term can be integrated individually, helping to manage the original complexity.

This technique extends beyond straightforward recognition of patterns to involve strategic manipulation of expressions to achieve a solvable state.