Evaluate the integrals in Exercises 1–22.
∫₀^π 3sin(x/3) dx

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Identify the integral to be evaluated: \(\int_0^{\pi} 3 \sin\left(\frac{x}{3}\right) \, dx\).

Use the substitution method by letting \(u = \frac{x}{3}\). Then, compute \(du = \frac{1}{3} dx\), which implies \(dx = 3 \, du\).

Change the limits of integration according to the substitution: when \(x = 0\), \(u = 0\); when \(x = \pi\), \(u = \frac{\pi}{3}\).

Rewrite the integral in terms of \(u\): \(\int_0^{\pi} 3 \sin\left(\frac{x}{3}\right) dx = \int_0^{\frac{\pi}{3}} 3 \sin(u) \cdot 3 \, du = \int_0^{\frac{\pi}{3}} 9 \sin(u) \, du\).

Integrate \(9 \sin(u)\) with respect to \(u\) using the integral formula \(\int \sin(u) \, du = -\cos(u) + C\), then substitute back the limits and evaluate the definite integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral calculates the net area under a curve between two specified limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating a definite integral involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Integration of Trigonometric Functions

Integrating trigonometric functions like sin(kx) requires using substitution or known integral formulas. For example, ∫ sin(kx) dx = -cos(kx)/k + C. Recognizing the coefficient inside the function is crucial for correctly adjusting the integral.

Substitution Method

The substitution method simplifies integrals by changing variables to make the integral easier to solve. For integrals like ∫ sin(x/3) dx, setting u = x/3 transforms the integral into a standard form, allowing straightforward integration and back-substitution.

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