Question

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int(x-\pi) \sin x d x $$

Short answer

\( \int (x - \pi) \sin x \, dx = -(x - \pi)\cos x + \sin x + C \).

Step-by-step solution

Identify Functions

For integration by parts, we need to identify two components in the integrand: one as a function to differentiate \( u \) and the other as a function to integrate \( dv \). Let's choose \( u = x - \pi \) and \( dv = \sin x \ dx \).

Differentiate and Integrate

Differentiate \( u \) and integrate \( dv \): \[ du = d(x - \pi) = dx \] \[ v = \int \sin x \, dx = -\cos x \]

Apply Integration by Parts Formula

Use the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \] Substitute \( u, v, du, \) and \( dv \) into the formula: \[ \int (x - \pi) \sin x \, dx = (x - \pi)(-\cos x) - \int -\cos x \, dx \]

Simplify the Expression

Simplify the expression: \[ = -(x - \pi)\cos x + \int \cos x \, dx \]

Evaluate the Remaining Integral

Evaluate the integral \( \int \cos x \, dx \): \[ = -(x - \pi)\cos x + \sin x + C \] Here, \( C \) is the constant of integration.

Key concepts

Understanding Definite Integrals

Definite integrals are a fundamental concept in calculus. They provide the area under a curve from one point to another along the x-axis, offering a precise calculation between two specified limits. Unlike indefinite integrals, which result in a general family of functions, definite integrals yield a specific numerical value.

• When you solve a definite integral, you're essentially summing infinitesimally small products of height (function value) and width (an increment in x) over an interval.
• This technique is useful for finding quantities such as areas, total accumulated change, and average values.

In practice, solving a definite integral involves evaluating the antiderivative at the upper limit and subtracting the evaluation at the lower limit. Remember, when working with definite integrals, the constant of integration typically cancels out because it appears in both evaluations of the limits.

Exploring Trigonometric Integration

Trigonometric integration refers to the process of integrating functions involving trigonometric expressions such as sine, cosine, and tangent.

• These integrals appear frequently in calculus problems, often requiring specific techniques or substitutions due to their periodic nature.
• Common strategies involve using trigonometric identities to simplify the integrand, making it easier to integrate.

When working with integrands like \( \sin x \) or \( \cos x \), you might need to apply identities such as \( \sin^2 x + \cos^2 x = 1 \) or substitution methods where you introduce a variable \( t \) that simplifies the integration process. The integral of \( \sin x \), for example, is \( -\cos x \), an important fact used when solving the original problem through integration by parts.

Approaching Calculus Problem Solving

Solving calculus problems, especially those involving integration, requires a structured approach:

• First, carefully analyze the problem and the integrand. This step helps in choosing the appropriate integration technique.
• Check for patterns or familiar structures in the integrand that suggest a specific method, such as substitution or integration by parts.
• Break the problem into smaller, more manageable parts if needed.

Understanding these steps leads to efficient problem-solving and avoids unnecessary complexity. For instance, in the integration problem \( \int(x- \pi) \sin x \, dx \), the choice of parts \( u \) and \( dv \) was crucial in simplifying and effectively solving the integral using integration by parts.

Mastering Integration Techniques

Various integration techniques are available in calculus to tackle different kinds of integrals. Familiarity with these techniques is key to solving integrals efficiently.

• Integration by Parts: This method is based on the product rule for differentiation and is particularly useful for integrals involving products of functions.
• Substitution: Often used to simplify integrands, this technique involves replacing a variable with a function of another variable.

The integration by parts formula, \( \int u \, dv = uv - \int v \, du \), facilitates the integration of products like \( \int(x- \pi) \sin x \, dx \). Mastering such techniques helps you navigate through complex integrals smoothly and solve problems more effectively. Practice frequently with these techniques to build confidence and proficiency.