Multiple substitutions If necessary, use two or more substitutions to find the following integrals.                                                                                    
                                                                                                                                                                    
  ∫ 𝓍 sin⁴ 𝓍² cos 𝓍² d𝓍 (Hint: Begin with u = 𝓍², and then use v = sin u .)

Verified step by step guidance

1. Start by identifying the first substitution as suggested: let \(u = x^{2}\). Then, compute the differential \(du = 2x \, dx\), which implies \(x \, dx = \frac{du}{2}\).
2. Rewrite the integral in terms of \(u\). Notice that \(\sin^{4}(x^{2}) = \sin^{4}(u)\) and \(\cos(x^{2}) = \cos(u)\). The integral becomes \(\int x \sin^{4}(x^{2}) \cos(x^{2}) \, dx = \int \sin^{4}(u) \cos(u) \cdot x \, dx = \int \sin^{4}(u) \cos(u) \cdot \frac{du}{2}\).
3. Simplify the integral to \(\frac{1}{2} \int \sin^{4}(u) \cos(u) \, du\). Now, observe that the integrand involves \(\sin^{4}(u)\) and \(\cos(u)\), which suggests a second substitution.
4. For the second substitution, let \(v = \sin(u)\). Then, \(dv = \cos(u) \, du\). This transforms the integral into \(\frac{1}{2} \int v^{4} \, dv\).
5. Now, the integral is a straightforward power integral in terms of \(v\). Integrate \(\frac{1}{2} \int v^{4} \, dv\) by increasing the power by one and dividing by the new exponent, then substitute back \(v = \sin(u)\) and \(u = x^{2}\) to express the answer in terms of \(x\).

Key Concepts

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves choosing a substitution u = g(x) such that the integral in terms of u is easier to evaluate. This technique is especially useful when the integral contains composite functions or products of functions.

Multiple Substitutions

Multiple substitutions involve applying the substitution method more than once to solve complex integrals. After the first substitution simplifies part of the integral, a second substitution can further reduce it. This stepwise approach is essential when a single substitution does not fully simplify the integral.

Trigonometric Functions and Their Derivatives

Understanding the derivatives and integrals of trigonometric functions like sine and cosine is crucial. For example, knowing that the derivative of sin u is cos u helps in choosing substitutions and simplifying integrals involving powers of sine and cosine. This knowledge aids in recognizing patterns and applying substitutions effectively.