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Antiderivative of f(x) = x² sin x³

This problem asks to find the antiderivative F(x) of the function f(x) = x² sin x³ satisfying the condition F(0) = 1/2.

Problem

f ( x ) = x2 sin x3 , F ( 0 ) = 12

Solution

Step 1

Note that the derivative of x³ is 3x². Therefore, d(x³) = 3x² dx. This allows us to bring the factor x² under the differential sign.

d ( x3 ) = 3 x2 d x

Step 2

Rewrite the original integral, extracting the factor 1/3:

∫ x2 sin x3 d x = 13 ∫ sin ( x3 ) d ( x3 )

Step 3

Perform the substitution t = x³. The integral then takes a tabular form:

13 ∫ sin t d t = − 13 cos t + C

Step 4

Return to the variable x:

F ( x ) = − 13 cos x3 + C

Step 5

Now find the constant C from the initial condition F(0) = 1/2. Substitute x = 0:

− 13 cos 0 + C = 12

Step 6

From this we get C = 5/6. The required antiderivative has the form:

Answer

F ( x ) = − 13 cos x3 + 56

Frequently asked questions

How do I find the antiderivative of f(x) = x² sin x³?

Use the substitution method. Note that d(x³) = 3x² dx, so the integral can be rewritten as (1/3) ∫ sin(x³) d(x³). Then apply the substitution t = x³, and the integral reduces to the tabular ∫ sin t dt.

Why is the constant C needed and how is it found?

The constant C appears in indefinite integration because the derivative of a constant is zero. Its value is determined from the initial condition — for example, F(0) = 1/2. Substitute x = 0 into the expression for F(x) and solve for C.

Why is the substitution t = x³ convenient in this problem?

The substitution t = x³ turns the difficult integral ∫ x² sin x³ dx into the tabular ∫ sin t dt. Without this substitution, the integral cannot be evaluated in elementary functions. The presence of the factor x², proportional to the derivative of x³, makes this substitution possible.

What is the difference between an antiderivative and an indefinite integral?

An indefinite integral is the family of all antiderivatives: ∫ f(x) dx = F(x) + C. An antiderivative with a specific value of C is one function from this family. The problem asks for the specific antiderivative satisfying the initial condition.

See also

  • Antiderivative and integral
  • Definite integral
  • Improper integrals
  • Double integrals
  • Triple integrals
  • Line integrals
  • Surface integrals
  • Elements of field theory
  1. Antiderivative of f(x) = x² sin x³.
  2. Antiderivative of f(x) = cos³ x.
  3. Antiderivative of f(x) = x · e^(3x).