如何求积分的原函数与导数？求解f’(x)及f(π)的方法问询

Hey there! Let's work through this together—first covering the basics of antiderivatives (原函数) and derivatives, then solving your specific integral equation problem.

• Derivatives and antiderivatives are inverse operations: If ( F(x) ) is an antiderivative of ( f(x) ), then ( F'(x) = f(x) ). Conversely, all antiderivatives of ( f(x) ) take the form ( F(x) + C ) (where ( C ) is any constant), which is exactly what the indefinite integral ( \int f(x)dx = F(x) + C ) represents.
• For definite integrals with variable upper limits, there's a key rule: If ( F(x) = \int_{a}^{x} f(t)dt ) (where ( a ) is a constant and ( f ) is continuous), then ( F'(x) = f(x) ). This is the rule that will unlock your problem!

First, let's restate your problem clearly:

Let ( f ) be a continuous function. For ( x > 0 ), ( f ) satisfies:
$$\int_{0}^{x}tf(t);dt = x\sin(x)+\cos(x)-1$$
Find ( f'(x) ) and ( f(\pi) ).

Step 1: Find ( f(x) ) #

We'll take the derivative of both sides of the equation with respect to ( x ):

• Left side: Using the variable upper limit integral rule, we replace ( t ) with ( x ) in the integrand ( tf(t) ). So the derivative is ( x f(x) ).
• Right side: Compute the derivative term by term:
  • Derivative of ( x\sin(x) ): Use the product rule: ( \sin(x) + x\cos(x) )
  • Derivative of ( \cos(x) ): ( -\sin(x) )
  • Derivative of ( -1 ): ( 0 )
  • Adding these together: ( (\sin(x) + x\cos(x)) - \sin(x) = x\cos(x) )

Now we have the equation:
$$x f(x) = x\cos(x)$$
Since ( x > 0 ), we can divide both sides by ( x ), giving us:
$$f(x) = \cos(x)$$

Step 2: Find ( f'(x) ) #

Now that we have ( f(x) = \cos(x) ), taking its derivative is straightforward:
$$f'(x) = -\sin(x)$$

Step 3: Find ( f(\pi) ) #

Substitute ( x = \pi ) into ( f(x) = \cos(x) ):
$$f(\pi) = \cos(\pi) = -1$$

A quick note: Your hunch about "extracting ( t )" was on the right track—this problem hinges on recognizing that we can use the variable upper limit integral rule to turn the integral equation into a simple algebraic equation for ( f(x) ).

内容的提问来源于stack exchange，提问作者deezy