int x^9*(sin(x^5))
Can someone please explain me how to do this integral?
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t=x^5, dt=5x^4dx
x^9*sin(x^5)dx=x^5*sin(x^5)*x^4*dx=
t*sin(t)*(1/5)dt
Int=1/5(sin(t)-t*sin(t))+C, t=x^5
Int=1/5(sin(t)-t*COS(t))+C
To integrate the expression x^9 * sin(x^5), we can follow these steps:
Step 1: Use substitution.
Let u = x^5. Then, differentiate both sides with respect to x to find du/dx = 5x^4.
Step 2: Solve for dx.
Rearrange the equation du/dx = 5x^4 to get dx = du/(5x^4).
Step 3: Rewrite the integral with the substitution.
Replace x^9 with (u^(9/5)) and dx with (du/(5x^4)).
The integral becomes ∫ (u^(9/5)) * sin(u) * (du/(5x^4)).
Step 4: Simplify the expression.
Since x^4 = (u^(4/5)), substitute (u^(4/5)) for x^4 in the denominator.
The integral simplifies to ∫ (u^(9/5)) * sin(u) * (du/(5(u^(4/5)))).
Step 5: Combine the terms.
The integral can be rewritten as (1/5) ∫ (u^(9/5 - 4/5)) * sin(u) du.
Simplifying further, we have (1/5) ∫ (u^1) * sin(u) du.
Step 6: Evaluate the integral.
Integrate (u^1) * sin(u) with respect to u. This can be done using integration by parts.
Let's use the following formula:
∫ u * sin(v) du = -u * cos(v) + ∫ cos(v) du
In this case, let u = u and dv = sin(u) du.
Then, du = 1 du and v = -cos(u).
Using the integration by parts formula, we find:
∫ (u^1) * sin(u) du = -u * cos(u) + ∫ cos(u) du.
The integral of cos(u) is sin(u) + C (where C is the constant of integration).
Step 7: Finalize the answer.
The final result is (1/5) * (-u * cos(u) + sin(u)) + C.
Replacing u with x^5, we have:
(1/5) * (-x^5 * cos(x^5) + sin(x^5)) + C.
Therefore, the integral of x^9 * sin(x^5) is (1/5) * (-x^5 * cos(x^5) + sin(x^5)) + C.