Q. 24 - Calculus

Question

Indefinite integrals: Use integration formulas, algebra, and educated guess-and-check strategies to find the following integrals.  

$\int x^3\sin \left(x^4\right)dx$

Step-by-Step Solution

Answer

The solution is:  $-\frac{1}{4}\cos \left(x^4\right)+C$

Step 1: Given

The given integral is:$\int x^3\sin \left(x^4\right)dx$

Step 2: To Find

Use integration formulas, algebra, and educated guess-and-check strategies to find the given integral. 

Step 3: Calculation

Let,

$$\begin{gathered} u=x^4 \\ du=4x^{3}dx \\ \frac{du}{4x^3}=dx \end{gathered}$$

Using substitution we get:

$$\begin{gathered} \int x^3\sin \left(x^4\right)dx \\ =\int x^3\sin \left(u\right)\frac{du}{4x^3} \\ =\frac{1}{4}\int \sin \left(u\right)du \\ =\frac{1}{4}\left[-\cos \left(u\right)\right]+C \\ =-\frac{1}{4}\cos \left(x^4\right)+C \end{gathered}$$

Where, C = integrating constant.