What's f(x)?

Question

If ∫ x^2*cos x dx=f(x)-∫ 2x sin x dx, what's f(x)?

 

Answer: x^2*sin x+C

 

 

Answer 1

You can do it by integration by parts mentally.

∫ x^2*cos x dx=f(x)-∫ 2x sin x dx ......(1)

 

∫ x^2*cos x dx

= ∫ x^2dsin x

= x^2 sin x - ∫ sin x dx^2 + C

 

=> ∫ x^2*cos x dx= x^2 sin x - ∫ 2x sin x dx ......(2)

 

(1)-(2): 0 = f(x) - x^2 sin x  - C

Therefore, f(x) = x^2 sin x + C

Answer 2

This requires integration by parts, which allows us to solve the integral of the product of two functions.

 

∫ x²cosx dx

 

This integral has two functions being multiplied together, x² and cosx. One of these we will assign to be itself, or u, and the other will be the derivative of a function v, or dv. We need to figure out which is which in order to continue.

 

A neat trick I use is the acronym LIPET, standing for Logarithmic, Inverse trigonometric, Polynomial, Exponential, Trigonometric. The order of this acronym tells you which function is the u and which function is the dv. We have x², which is a polynomial function, and cosx, which is a trigonometric. P comes before T in the acronym, so x² is the u and cosx is the dv.

 

The rule for integration by parts is: ∫ u dv = uv - ∫ v du. We now know u and dv, so we still need to find du and v, by differentiating u and integrating dv. This is why we use the LIPET to assign u and dv, since it allows us to find functions that we can easily differentiate and integrate. 

 

u = x²               v = sinx

du = 2x dx        dv = cosx dx 

 

uv - ∫ v du = (x²)(sinx) - ∫ sinx(2x dx) = x²sinx - ∫ 2xsinx dx + C

 

Based on your question, the f(x) you're looking for is x²sinx + C.

 

If you were to continue to solve this integral, you would need to use integration by parts again for ∫ 2xsinx dx. Again, 2x is a polynomial, and sinx is trigonometric, so 2x would be the u and sinx would be the dv.

Answer 3

Rewrite as f(x) = ∫ (2x sin x + x² cos x) dx

 

Note that the integrand has the form d/dx (uv) = u'v+uv' where u(x) = x² and v(x) = sin x.

 

Hence f(x) = uv = x² sin x + C