Integrate:(3x^5-5)^-3 dx
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To integrate the given expression, which is ∫(3x^5-5)^-3 dx, we can use a substitution method. Let's do the following substitution:
Let u = 3x^5 - 5
Now we need to find the derivative of u with respect to x:
du/dx = d/dx (3x^5 - 5)
du/dx = 15x^4
We can rearrange this equation to solve for dx:
dx = du / (15x^4)
Now we can substitute these values back into the integral:
∫(3x^5-5)^-3 dx = ∫(u)^-3 * (du / (15x^4))
Now our integral becomes:
∫(u)^-3 * (du / (15x^4))
Next, we can simplify the expression by using exponent rules.
(u)^-3 = 1 / (u)^3
Substituting this back into the integral:
∫(1 / (u)^3) * (du / (15x^4))
Since u = 3x^5 - 5, we can substitute u back into the integral:
∫(1 / (3x^5 - 5)^3) * (du / (15x^4))
Now, we can simplify further:
∫(1 / 15x^4 * (3x^5 - 5)^3) du
We can simplify the expression by multiplying the constants:
1 / 15 * ∫(x^4 * (3x^5 - 5)^3) du
Now, we can distribute the x^4 through the (3x^5 - 5)^3 term:
1 / 15 * ∫(3x^9 - 15x^4) du
Integrating each term separately:
(1 / 15) * [∫(3x^9) du - ∫(15x^4) du]
Now, we integrate each term:
= (1 / 15) * [(u^10 / 10) - (15u^5 / 5)] + C
= (1/ 15) * [(u^10 / 10) - 3u^5] + C
Finally, we substitute u back into the expression:
= (1 / 15) * [(3x^5 - 5)^10 / 10 - 3(3x^5 - 5)^5] + C
This is the expression for the integral ∫(3x^5-5)^-3 dx.