Integration by Parts: Calculating the Antiderivative of x2sin(x)

Integration by parts is a powerful technique used in calculus to integrate the product of two functions. The formula for integration by parts is given by:

Integration by Parts Formula: ( int u , dv uv - int v , du )

In this article, we will use the specific case of finding the antiderivative of (x^2 sin(x)). Let's dive into the step-by-step process of applying integration by parts.

Step-by-Step Integration of x2sin(x)

First Application of Integration by Parts: Choose (u x^2) and (dv sin(x) , dx). Then, (du 2x , dx) and (v -cos(x)). Applying the formula:

(int x^2 sin(x) , dx -x^2 cos(x) - int 2x cos(x) , dx)

In the second integral, we apply integration by parts again:

Second Application of Integration by Parts: Choose (u 2x) and (dv cos(x) , dx). Then, (du 2 , dx) and (v sin(x)). Applying the formula again:

(int 2x cos(x) , dx 2x sin(x) - int 2 sin(x) , dx)

Now, we integrate (int 2 sin(x) , dx):

(int 2 sin(x) , dx -2 cos(x) )

Putting it all together:

(int x^2 sin(x) , dx -x^2 cos(x) 2x sin(x) 2 cos(x) C)

Understanding Antiderivatives

The antiderivative of a function is the integral of that function. For example, the antiderivative of (x^2) is (frac{x^3}{3}), not (sin(x)). Equating (x^2) to (sin(x)) as the antiderivative would be incorrect, as the derivative of (sin(x)) is (cos(x)), not (x^2).

Example: Integral of sin(2x)

Another example to demonstrate the application of integration by parts:

(int sin(2x) , dx -frac{1}{2} cos(2x) C)

Conclusion

Integration by parts is a crucial technique for integrating complex functions. Understanding the steps and applying the formula consistently can help in finding the antiderivative accurately. Whether it's (x^2 sin(x)), (sin(2x)), or other functions, the key is to carefully choose (u) and (dv) and apply the integration by parts formula.

[Keyword]: antiderivative, integration by parts, trigonometric functions