Question

62. Two integration methods Evaluate ∫ sin x cos x dx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers

Accepted Answer

First, let's evaluate the integral $$\int \sin x \cos x \, dx$$ using integration by parts. Recall the formula for integration by parts: $$\int u \, dv = uv - \int v \, du. We $$need to choose $$u$$ and $$dv$$ from the integrand. Choose $$u = \sin x$$ and $$dv = \cos x \, dx. $$Then compute $$du = \cos x \, dx$$ and $$v = \int \cos x \, dx = \sin x. $$Substitute these into the integration by parts formula: $$\int \sin x \cos x \, dx = \sin x \cdot \sin x - \int \sin x \cdot \cos x \, dx = \sin^2 x - \int \sin x \cos x \, dx.$$ Notice that the integral $$\int \sin x \cos x \, dx$$ appears on both sides of the equation. Let's call this integral $$I. $$Then we have $$I = \sin^2$ x - I$$, which can be rearranged to solve for $$I. $$Next, evaluate the integral using substitution. Let $$t = \sin x$$, so that $$dt = \cos x \, dx. $$The integral becomes $$\int t \, dt$$, which is straightforward to integrate. After integrating, substitute back $$t = \sin x to $$express the answer in terms of $$x. $$Finally, compare this result with the one obtained from integration by parts to reconcile any differences, such as constant terms.

62. Two integration methods Evaluate ∫ sin x cos x dx using integration by parts. Then evaluate the integral using a substitution. Reconcile your answers

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Key Concepts

Integration by Parts

Substitution Method

Reconciling Different Integration Results