Evaluate $\displaystyle \int{3x^2\sin{\big(x^3\big)}\,}dx$

$x$ squared and $x$ cubed are two power functions but the cube of $x$ represents an angle inside the sine function. The indefinite integral of three times the product of $x$ squared and sine of angle $x$ cubed with respect to $x$ is written as follows.

$\displaystyle \int{3x^2\sin{\big(x^3\big)}\,}dx$

Let’s learn how to find the indefinite integration of three times product of square of $x$ and sine of angle cube of $x$ with respect to $x$.

Minimize the Complexity of the function

In this indefinite integral problem, the given function is a combination and composition of both algebraic and trigonometric functions. Their complex formation creates problem for the integration. So, it is better to think about converting the function into simple form.

$x^3$ is a power function and it represents an angle inside the sine function but its derivative is also involved as a factor in the given function. So, it can be simplified by representing the power function with a variable.

Assume $u \,=\, x^3$

Now, differentiate the expressions on both sides of the equation with respect to $x$.

$\implies$ $\dfrac{d}{dx}{(u)} \,=\, \dfrac{d}{dx}{\big(x^3\big)}$

The derivative of power function can be evaluated by the power rule of derivatives.

$\implies$ $\dfrac{du}{dx} \,=\, 3 \times x^{3-1}$

$\implies$ $\dfrac{du}{dx} \,=\, 3 \times x^2$

$\implies$ $\dfrac{du}{dx} \,=\, 3x^2$

$\implies$ $du \,=\, 3x^2 \times dx$

$\,\,\,\therefore\,\,\,\,\,\,$ $du \,=\, 3x^2 dx$

Transform the function in simple form

We have taken that $u \,=\, x^3$ and derived that $du \,=\, 3x^2 dx$. Now, let us convert the given integral function in terms of u by using these two mathematical relations.

$=\,\,\,$ $\displaystyle \int{3x^2 \times \sin{\big(x^3\big)}\,} \times dx$

$=\,\,\,$ $\displaystyle \int{\sin{\big(x^3\big)} \times 3x^2 \times\,}dx$

$=\,\,\,$ $\displaystyle \int{\sin{\big(x^3\big)} \times 3x^2\,}dx$

$\implies$ $\displaystyle \int{\sin{\big(x^3\big)} \times 3x^2\,}dx$ $\,=\,$ $\displaystyle \int{\sin{(u)}}du$

Find the integral of the function

The given integral function in $x$ is successfully simplified as follows in terms of a variable $u$.

$\displaystyle \int{\sin{u}\,}du$

The integral of sine of angle $u$ with respect to $u$ can be calculated as per the integral rule of sine function.

$\,\,\,=\,$ $-\cos{u}+c$

The variable $u$ is taken to represent the power function $x$ cube. So, it is time to bring back the function in terms of $x$ by replacing the value of $u$.

$\,\,\,=\,$ $-\cos{\big(x^3\big)}+c$