Question

Use the Substitution Rule to prove that $$\begin{array}{l}\int {\sin }^{2}axdx=\frac{x}{2}-\frac{\sin (2ax)}{4a}+C\text{ and }\\ \int {\cos }^{2}axdx=\frac{x}{2}+\frac{\sin (2ax)}{4a}+C\end{array}$$

Short answer

Question: Show how to integrate ${\sin }^2(ax)$ and ${\cos }^2(ax)$. Answer: To integrate these functions, we can use the trigonometric double-angle identities along with substitution. The results are: $\int {\sin }^2(ax)dx=\frac{x}{2}-\frac{\sin (2ax)}{4a}+C$ $\int {\cos }^2(ax)dx=\frac{x}{2}+\frac{\sin (2ax)}{4a}+C$

Step-by-step solution

Write down the trigonometric identities

Using the trigonometric identities for double angles, rewrite the given integrals. $$\int {\sin }^2(ax)dx=\int \frac{1-\cos (2ax)}{2}dx$$ $$\int {\cos }^2(ax)dx=\int \frac{1+\cos (2ax)}{2}dx$$

Integration of ${\sin }^2(ax)$

Now, we need to integrate ${\sin }^2(ax)$ using the Substitution Rule. Let's first set the substitution: $$u=2ax$$ $$du=2adx$$ $$\frac{1}{2a}du=dx$$ And apply this substitution to the integral: $$\int \frac{1-\cos (2ax)}{2}dx=\frac{1}{2}\int (1-\cos (u))\frac{1}{2a}du$$ $$=\frac{1}{4a}\int (1-\cos (u))du$$ Now, integrate with respect to $u$: $$\frac{1}{4a}[\int 1du-\int \cos (u)du]=\frac{1}{4a}[u-\sin (u)]+C$$ Finally, revert the substitution $u=2ax$: $$\frac{1}{4a}[2ax-\sin (2ax)]+C=\frac{x}{2}-\frac{\sin (2ax)}{4a}+C$$

Integration of ${\cos }^2(ax)$

Similarly, we need to integrate ${\cos }^2(ax)$ using the Substitution Rule. We already have the substitution from before: $$u=2ax$$ $$du=2adx$$ $$\frac{1}{2a}du=dx$$ Apply this substitution to the integral: $$\int \frac{1+\cos (2ax)}{2}dx=\frac{1}{2}\int (1+\cos (u))\frac{1}{2a}du$$ $$=\frac{1}{4a}\int (1+\cos (u))du$$ Now, integrate with respect to $u$: $$\frac{1}{4a}[\int 1du+\int \cos (u)du]=\frac{1}{4a}[u+\sin (u)]+C$$ Finally, revert the substitution $u=2ax$: $$\frac{1}{4a}[2ax+\sin (2ax)]+C=\frac{x}{2}+\frac{\sin (2ax)}{4a}+C$$ Thus, we have proven the given expressions: $$\int {\sin }^2(ax)dx=\frac{x}{2}-\frac{\sin (2ax)}{4a}+C$$ $$\int {\cos }^2(ax)dx=\frac{x}{2}+\frac{\sin (2ax)}{4a}+C$$

Key concepts

Integrals of Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in mathematics, especially when dealing with periodic phenomena. Integrating these functions, however, often requires more than a basic understanding of integration. With the Substitution Rule, we can transform the integral of a trigonometric function into a simpler form that can be more easily integrated.

For example, when integrating the square of sine or cosine, we use the known trigonometric identities to rewrite the integrals in terms of a double angle. This technique simplifies the integrand, allowing us to apply substitution effectively. With the substitution $u=2ax$, we can deal with a linear term and a simpler trigonometric function, paving the way for straightforward integration and arriving at the solution.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. These identities are indispensable tools in calculus, especially in integration and differentiation of trigonometric functions.

For instance, the double angle identities for sine and cosine, $\sin (2x)=2\sin (x)\cos (x)$ and $\cos (2x)={\cos }^2(x)-{\sin }^2(x)$, are instrumental when tackling integrals involving squared trigonometric functions. By expressing ${\sin }^2(x)$ and ${\cos }^2(x)$ in terms of double angles, we can simplify the integrals, as shown in the exercise above, which greatly assists in the integration process by substituting and integrating more manageable functions.

Integration Techniques

There are several techniques used to integrate more complex functions, and choosing the right one is crucial to solving an integration problem. For trigonometric functions, common techniques include substitution, integration by parts, and trigonometric identities.

In the provided exercise, the Substitution Rule is employed to integrate functions that would otherwise be more difficult to handle. This technique involves replacing a part of the integrand with a new variable, making the integral simpler. After integration, the substitution is reverted back to the original variable. Users should also be aware that after simplifying the trigonometric integrals using identities, the integrals often break down into basic forms that are straightforward to integrate, demonstrating the power of combining these techniques.