Welcome to this integration tutorial. Today we will solve the integral of x squared sine x using the integration by parts technique. This is a classic example that requires applying integration by parts twice to reach the final solution.
Now let's apply the integration by parts formula. The formula states that the integral of u dv equals u times v minus the integral of v du. For our integral, we choose u equals x squared and dv equals sine x dx. Using the LIATE rule, we pick the algebraic term x squared as u since algebraic functions have higher priority than trigonometric functions. Then we find du equals 2x dx and v equals negative cosine x.
Now let's apply the integration by parts formula. We substitute our values: u equals x squared, v equals negative cosine x, du equals 2x dx. This gives us negative x squared cosine x minus the integral of negative cosine x times 2x dx. Simplifying the signs, we get negative x squared cosine x plus the integral of 2x cosine x dx. We can factor out the 2 to get negative x squared cosine x plus 2 times the integral of x cosine x dx. Notice that we still have an integral that requires integration by parts again.
Now we solve the remaining integral x cosine x dx using integration by parts again. We choose u equals x and dv equals cosine x dx. Then du equals dx and v equals sine x. Applying the formula gives us x sine x minus the integral of sine x dx. The integral of sine x is negative cosine x, so we get x sine x minus negative cosine x, which simplifies to x sine x plus cosine x.
Finally, let's substitute our result back into the original expression. We had negative x squared cosine x plus 2 times the integral of x cosine x dx. We found that the integral of x cosine x dx equals x sine x plus cosine x. Substituting this back gives us negative x squared cosine x plus 2 times the quantity x sine x plus cosine x. Expanding this, we get negative x squared cosine x plus 2x sine x plus 2 cosine x. Adding the constant of integration, our final answer is negative x squared cosine x plus 2x sine x plus 2 cosine x plus C.