Answer to Question #99241 in Calculus for Leo

Question

A complex function can be modified by the equation: y = cos(x^3) 3x^2
Find the indefinite integral of the function ( ∫ cos(x^3)3x^2 dx) using a substitution method?

Answer 1

^{Indefinite Integral of a function}

^{We need to find the Indefinite Integration of a function using a Substitution method.}

^Solution:

^Given,

$$\int cos (x^3 ) \space 3x^2 \space dx$$

$$Let \space x^3 = t \space\\ then \space d (x^3 ) = dt$$

(since , d (xⁿ ) = n . x ^{(n - 1)} dx)

$$3 \times x^2 dx = dt$$

Plug all these in the given Integral, then we get

$$\int cos (x^3 ) \space 3x^2 \space dx = \int cos t \space dt$$

$$= \int cos t \space dt = sint + C$$

re-plug the substitution,

$$\int cos (x^3 ) \space 3x^2 \space dx = sin \space t + C = sin \space x^3 + C$$

Answer:

$$\int cos (x^3 ) \space 3x^2 \space dx = sin \space x^3 + C$$