Calculate First Arch Integral of Xsinx

The first arch integral of xsinx refers to the antiderivative of the function xsinx. This calculation is fundamental in calculus and has applications in physics, engineering, and other scientific fields. Our calculator provides an accurate and efficient way to compute this integral.

What is the First Arch Integral?

The first arch integral, also known as the indefinite integral, is the antiderivative of a function. For the function xsinx, the first arch integral represents the area under the curve of xsinx from some starting point to x. This concept is crucial in calculus for solving differential equations and finding areas under curves.

In practical terms, the first arch integral of xsinx helps in determining the total accumulation of xsinx over a given interval. This is valuable in physics for calculating work done by variable forces, in engineering for analyzing systems with changing rates, and in economics for modeling continuous growth.

Formula for First Arch Integral

The formula for the first arch integral of xsinx is derived using integration by parts. The general formula is:

∫x sinx dx = -x cosx + ∫cosx dx

This can be further simplified to:

∫x sinx dx = -x cosx + sinx + C

where C is the constant of integration. This formula allows us to compute the antiderivative of xsinx accurately.

How to Calculate First Arch Integral

Calculating the first arch integral of xsinx involves applying integration by parts. Here are the steps:

Identify the function to integrate: xsinx.
Apply the integration by parts formula: ∫u dv = uv - ∫v du.
Let u = x and dv = sinx dx.
Compute du = dx and v = -cosx.
Substitute into the integration by parts formula: ∫x sinx dx = -x cosx + ∫cosx dx.
Integrate cosx to get sinx.
Combine the results to get the final antiderivative: -x cosx + sinx + C.

Note: The constant of integration C is added to represent the infinite family of solutions for the indefinite integral.

Example Calculation

Let's compute the first arch integral of xsinx from 0 to π. Using the formula:

∫₀^π x sinx dx = [-x cosx + sinx]₀^π

Evaluating at the bounds:

At x = π: -π cosπ + sinπ = -π (-1) + 0 = π
At x = 0: -0 cos0 + sin0 = 0 + 0 = 0

Subtracting the lower bound from the upper bound gives the definite integral:

π - 0 = π

This means the area under the curve of xsinx from 0 to π is π.

FAQ

What is the difference between definite and indefinite integrals?

An indefinite integral represents a family of functions that differ by a constant, while a definite integral computes the exact area under the curve between specified bounds.

Why is integration by parts used for xsinx?

Integration by parts is used when the function is a product of two functions, like x and sinx, and it simplifies the integration process by breaking it into more manageable parts.

How can I verify the result of the first arch integral?

You can verify the result by differentiating the antiderivative and checking if you get back to the original function. For xsinx, differentiating -x cosx + sinx + C should yield xsinx.