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 .)

검증된 단계별 안내

Start by identifying the first substitution as suggested: let \(u = x^{2}\). Then, compute the differential \(du = 2x \, dx\), which implies \(x \, dx = \frac{du}{2}\).

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 \(\int x \sin^{4}(x^{2}) \cos(x^{2}) \, dx = \int \sin^{4}(u) \cos(u) \cdot x \, dx = \int \sin^{4}(u) \cos(u) \cdot \frac{du}{2}\).

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.

For the second substitution, let \(v = \sin(u)\). Then, \(dv = \cos(u) \, du\). This transforms the integral into \(\frac{1}{2} \int v^{4} \, dv\).

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 \(x\).

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주요 개념

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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.