Solving Integrals A Step-by-Step Guide To ∫(cos²(2x) - Sin²(2x))³sin(4x) Dx And More

4.1.1 ∫(cos²(2x) - sin²(2x))³sin(4x) dx

In this section, we will delve into the solution of the integral ∫(cos²(2x) - sin²(2x))³sin(4x) dx. This integral appears complex at first glance, but by employing trigonometric identities and a clever substitution, we can simplify it into a manageable form. The key lies in recognizing the double-angle formula for cosine, which states that cos(2θ) = cos²(θ) - sin²(θ). Applying this identity to our integral, we can rewrite the expression inside the parentheses.

Trigonometric Identities and Simplification

Our integral involves the term (cos²(2x) - sin²(2x))³, which, as mentioned earlier, is closely related to the double-angle formula for cosine. By substituting cos(4x) for (cos²(2x) - sin²(2x)), we significantly simplify the expression. This substitution transforms our integral into ∫(cos(4x))³sin(4x) dx, which is much easier to handle. This step highlights the importance of recognizing and applying appropriate trigonometric identities to simplify integrals.

U-Substitution

Now that we have simplified the integral, we can employ the u-substitution method. This technique involves substituting a part of the integrand with a new variable, 'u', and its derivative. In our case, a suitable substitution is u = cos(4x). Differentiating both sides with respect to x, we get du = -4sin(4x) dx. This substitution allows us to rewrite the integral in terms of 'u', making it easier to integrate. The power of u-substitution lies in its ability to transform complex integrals into simpler forms, often involving basic power rules of integration.

Evaluating the Integral in Terms of u

After substituting u = cos(4x) and du = -4sin(4x) dx, our integral becomes -1/4 ∫u³ du. This integral is a straightforward application of the power rule for integration, which states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, where C is the constant of integration. Applying this rule, we find that -1/4 ∫u³ du = -1/16 u⁴ + C. This step demonstrates the simplicity that u-substitution can bring to integral evaluation.

Back-Substitution and Final Result

Having evaluated the integral in terms of 'u', we need to substitute back to express the result in terms of the original variable, 'x'. Replacing 'u' with cos(4x), we obtain the final result: -1/16 cos⁴(4x) + C. This is the solution to the integral ∫(cos²(2x) - sin²(2x))³sin(4x) dx. The entire process showcases a combination of trigonometric identities and substitution techniques to solve a seemingly complex integral.

4.1.2 ∫(1/x³)(1 + 1/x²)³ dx

This integral, ∫(1/x³)(1 + 1/x²)³ dx, presents a different challenge compared to the previous one. It involves rational functions and a power of a binomial. To tackle this, we will again use the u-substitution method, but with a different choice of 'u'. The key here is to identify a suitable expression within the integrand whose derivative is also present, allowing for simplification.

Strategic U-Substitution

Looking at the integral, we notice that the expression (1 + 1/x²) appears raised to the power of 3. This suggests that we might benefit from substituting u = (1 + 1/x²). The derivative of 'u' with respect to 'x' is du/dx = -2/x³, which is conveniently present in the integral (apart from a constant factor). This confirms that our choice of 'u' is likely to lead to a simplification. Selecting the right substitution is crucial for effectively using this technique.

Rewriting the Integral in Terms of u

Substituting u = (1 + 1/x²), we have du = -2/x³ dx. To match the original integral, we can rewrite this as -1/2 du = 1/x³ dx. Now we can substitute both (1 + 1/x²) and 1/x³ dx in the original integral. This transforms the integral into -1/2 ∫u³ du. This transformation is a prime example of how u-substitution simplifies complex integrals into manageable forms.

Applying the Power Rule of Integration

The integral -1/2 ∫u³ du is a straightforward application of the power rule for integration. As we saw earlier, the power rule states that ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C. Applying this rule to our integral, we get -1/2 ∫u³ du = -1/2 * (u⁴/4) + C = -1/8 u⁴ + C. This step highlights the ease with which the power rule can be applied after a suitable substitution.

Back-Substitution and Final Solution

Finally, we substitute back to express the result in terms of the original variable, 'x'. Replacing 'u' with (1 + 1/x²), we obtain the final solution: -1/8 (1 + 1/x²)⁴ + C. This is the result of the integral ∫(1/x³)(1 + 1/x²)³ dx. The solution process emphasizes the strategic use of u-substitution and the power rule to solve integrals involving rational functions.

4.1.3 ∫sin(3x/2)sin(5x/2) dx

This integral, ∫sin(3x/2)sin(5x/2) dx, involves the product of two sine functions with different arguments. To solve this, we'll employ a product-to-sum trigonometric identity. These identities are crucial for simplifying integrals involving products of trigonometric functions. Recognizing the appropriate identity is key to solving this type of integral.

Product-to-Sum Identity

The relevant product-to-sum identity for our integral is: sin(A)sin(B) = 1/2[cos(A - B) - cos(A + B)]. This identity allows us to rewrite the product of two sine functions as a difference of two cosine functions. In our case, A = 3x/2 and B = 5x/2. Applying this identity is a fundamental step in simplifying the integral.

Applying the Identity to the Integral

Substituting A = 3x/2 and B = 5x/2 into the product-to-sum identity, we get: sin(3x/2)sin(5x/2) = 1/2[cos(3x/2 - 5x/2) - cos(3x/2 + 5x/2)] = 1/2[cos(-x) - cos(4x)]. Since cos(-x) = cos(x), we can further simplify this to 1/2[cos(x) - cos(4x)]. This transformation simplifies the integral significantly, making it easier to evaluate.

Integrating the Simplified Expression

Now our integral becomes 1/2 ∫[cos(x) - cos(4x)] dx. This integral can be split into two simpler integrals: 1/2 ∫cos(x) dx - 1/2 ∫cos(4x) dx. The integral of cos(x) is sin(x), and the integral of cos(4x) is (1/4)sin(4x). Therefore, the integral becomes 1/2[sin(x) - 1/4 sin(4x)] + C. This step demonstrates the linearity of integration and the ease of integrating cosine functions.

Final Solution

Simplifying the expression, we get the final solution: 1/2 sin(x) - 1/8 sin(4x) + C. This is the result of the integral ∫sin(3x/2)sin(5x/2) dx. The solution process highlights the importance of using trigonometric identities to simplify integrals involving products of trigonometric functions.

In conclusion, these examples illustrate the diverse techniques used in solving integrals. From u-substitution to trigonometric identities, a combination of methods is often required to tackle complex integrals. Understanding these techniques is crucial for mastering integral calculus.