# How Can You Integrate xsinxcosxdx Using Exponential Form?

• carlosgrahm
In summary, to integrate xsinxcosxdx, use integration by parts with u=x and dv=sinxcosxdx. This will result in x(sinx)2/2 and the integral of (1/2)(sinx)2dx. You can then use the double angle formula for cos to get rid of (sinx)2/2 and rewrite the integrand as x/2 * sin(2x). By making the substitution y=2x and using integration by parts, you can easily integrate y/4 * sin(y). This method can be applied to solve any similar problems by expressing trigonometric functions in terms of their exponential form and multiplying everything out.
carlosgrahm
How do you integrate

xsinxcosxdx

Inegrate by parts: u=x, dv=sinxcosxdx=sinxd(sinx)
You get x(sinx)2/2 -integral of (1/2)(sinx)2dx

You should be able to proceed (using double angle formula for cos to get rid of (sinx)2/2).

Since
2sin(x)cos(x) = sin(2x)
you can write the integrand function
$$x/2 \cdot \sin (2x)$$
you can use first the substitution
$$y=2x$$
and then use integration by part formula to integrate
$$y/4 \cdot \sin (y)$$
it is EASY if you choose to derive $$y/4$$ and integrate $$\sin(y)$$.

You can solve any question like this by expressing sin(x), cos(x), etc in terms of their exponential form and multiplying everything out.

cos(x) = [exp(ix)+exp(-ix)]/2
sin(x) = [exp(ix)-exp(-ix)]/(2i)

## 1. What is the formula for integrating xsinxcosxdx?

The formula for integrating xsinxcosxdx is ∫xsinxcosxdx = ½ [x^2sin2x - ∫sin2xdx].

## 2. How do you solve the integral of xsinxcosxdx?

To solve the integral of xsinxcosxdx, you can use the substitution method by letting u = sinx and du = cosxdx. This will result in the integral becoming ∫xsinxcosxdx = ∫usinu du. You can then use integration by parts to solve the remaining integral.

## 3. Can you simplify the integral of xsinxcosxdx further?

No, the integral of xsinxcosxdx cannot be simplified further. It is considered a "trigonometric" integral and does not have a simple solution like other integrals.

## 4. What are the limits of integration for the integral of xsinxcosxdx?

The limits of integration for the integral of xsinxcosxdx depend on the specific problem given. Typically, you will be given specific values for x or u to plug into the integral after solving it.

## 5. Is there a specific method or trick to solve the integral of xsinxcosxdx?

Yes, the substitution method and integration by parts are the most commonly used methods to solve the integral of xsinxcosxdx. However, the specific method used may vary depending on the given problem.

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