Evaluate the Integral
1. integral of (x^9+7x^6-1)/x^8 dx
2. integral of x^(1/3)*(42-x)^2 dx
3. integral of 9x+5/7x^3 dx
To evaluate each integral, we will apply basic integration techniques. Let's go through each integral step by step:
1. integral of (x^9+7x^6-1)/x^8 dx:
To simplify this integral, we can split it into three separate integrals:
∫(x^9/x^8) dx + ∫(7x^6/x^8) dx - ∫(1/x^8) dx
Simplifying each integral separately:
∫(x^1) dx + 7∫(x^-2) dx - ∫(x^-8) dx
Now, apply the power rule of integration:
(x^(1+1))/(1+1) + 7(x^-2+1)/-2+1 -(x^-8+1)/-8+1
Simplifying further:
(x^2)/2 - 7(x^-1)/2 + (x^-7)/7 + C
2. integral of x^(1/3)*(42-x)^2 dx:
To integrate this expression, we'll use the power rule and the binomial expansion.
Start by expanding the expression (42-x)^2:
(42-x)^2 = 42^2 - 2(42)(x) + x^2
= 1764 - 84x + x^2
Now, the original integral becomes:
∫x^(1/3) * (1764 - 84x + x^2) dx
Distribute the x^(1/3) to each term:
∫(x^(1/3) * 1764) dx - ∫(x^(1/3) * 84x) dx + ∫(x^(1/3) * x^2) dx
Apply the power rule to each term:
(1764/4)x^(1/3+1) - (84/3)x^(1/3+2) + (1/3)x^(1/3+3) + C
Simplifying further:
441x^(4/3) - 28x^(7/3) + (1/3)x^(10/3) + C
3. integral of (9x+5)/(7x^3) dx:
To solve this integral, we notice that we have a polynomial divided by a function of x. We can use the method of u-substitution to simplify the integration.
Let's start by setting u equal to the denominator:
u = 7x^3
Taking the derivative of u with respect to x:
du/dx = 21x^2
Rearranging the equation, we have:
dx = du/(21x^2)
Substituting u and dx into the original integral, we get:
∫(9x+5)/(7x^3) dx = ∫(9x+5)/(u) (du/(21x^2))
Simplifying:
∫(9x+5)/(7x^3) dx = (1/21) ∫(9x+5)/u du
Now, we can integrate with respect to u:
(1/21) ∫(9x+5)/u du = (1/21) (9∫(1/u) du + 5∫(1/u) du)
Evaluating the integrals:
(1/21) (9ln|u| + 5ln|u|) + C
Substituting back u = 7x^3:
(1/21) (9ln|7x^3| + 5ln|7x^3|) + C
Simplifying further:
(14/7)ln|7x^3| + C
2ln|7x^3| + C