Question

Verify the following indefinite integrals by differentiation. These integrals are derived in later chapters. $$\int x^{2} \cos x^{3} d x=\frac{1}{3} \sin x^{3}+C$$

Short answer

Question: Verify the indefinite integral: $$\int x^{2} \cos x^{3} dx = \frac{1}{3} \sin x^{3} + C$$ Answer: The indefinite integral is verified since the derivative of the antiderivative is equal to the original integrand: $$x^{2}\cos x^{3}$$

Step-by-step solution

Identify the antiderivative and the integrand

The given indefinite integral is: $$\int x^{2} \cos x^{3} dx$$ The claimed antiderivative is: $$\frac{1}{3} \sin x^{3} + C$$ where C is the constant of integration.

Apply the chain rule to differentiate the antiderivative

To verify the given indefinite integral, we will differentiate the claimed antiderivative: $$\frac{d}{dx} \left(\frac{1}{3} \sin x^{3} + C\right)$$ Recall the chain rule: If \(u(x)=a(x) \cdot b(x)\), then \(u'(x) = a'(x) \cdot b(x)+a(x) \cdot b'(x)\) Using the chain rule, we have: $$\frac{d}{dx} \left(\frac{1}{3} \sin x^{3} + C\right) = \frac{1}{3}(3x^{2}\cos x^{3}) = x^{2}\cos x^{3}$$

Compare the obtained derivative with the original integrand

The derivative obtained in step 2 is: $$x^{2}\cos x^{3}$$ The original integrand is: $$x^{2}\cos x^{3}$$ As both the original integrand and the derivative of the antiderivative are equal, the given indefinite integral is verified.