int x^9*(sin(x^5))
Can someone please explain me how to do this integral?
To integrate the function \(x^9 \cdot \sin(x^5)\), we can use a technique called substitution. This involves substituting a new variable for a given expression in order to simplify the integral.
Here's a step-by-step guide on how to do this integral:
Step 1: Let \(u = x^5\). This substitution will help us simplify the integral.
Step 2: Find the derivative \(du\) with respect to \(x\) by differentiating both sides of the equation \(u = x^5\) with respect to \(x\).
- \(du = 5x^4 dx\)
Step 3: Rearrange the expression to solve for \(dx\).
- \(dx = \frac{du}{5x^4}\)
Step 4: Substitute the new variables and their differentials into the original integral.
- \(\int x^9 \cdot \sin(x^5) dx = \int \frac{1}{5} x^4 \cdot 5x^4 \cdot \sin(u) du\)
- Simplifying, we have \( \frac{1}{5} \int (x^4)^2 \cdot \sin(u) du\)
Step 5: Simplify the expression inside the integral:
- \(\frac{1}{5} \int x^8 \cdot \sin(u) du\)
Step 6: Integrate the simplified expression:
- \(\frac{1}{5} \int x^8 \cdot \sin(u) du = -\frac{1}{5} \cos(u) + C\)
where \(C\) is the constant of integration.
Step 7: Replace \(u\) with the original expression \(x^5\):
- \(-\frac{1}{5} \cos(x^5) + C\)
That's it! The integral of \(x^9 \cdot \sin(x^5)\) is \(-\frac{1}{5} \cos(x^5) + C\), where \(C\) is the constant of integration.