What is the answer for these questions:-
1) Indefinite Integrals gcx) = (8 + 39x ^ 3) / x
2) Indefinite Integrals hcu) = sin ^2 (1/8 u)
3) Evaluate x ( 8 - 5 x ^2) dx
Thank you
1) What does the gcx mean?
Treat the integrand as 8/x + 39 x^2
The indefinite integral of that is
8 lnx +13 x^3
2) What does the hcu mean?
3) What are the upper and lower limits of x?
1) To find the indefinite integral of the function gc(x) = (8 + 39x^3) / x, we can use the power rule for integration. The power rule states that the integral of x^n with respect to x is (x^(n+1))/(n+1), for any n not equal to -1.
In this case, we have gc(x) = (8 + 39x^3) / x. To integrate this function, we can separate it into two terms: 8/x + 39x^3/x.
For the first term, 8/x, we can see that the power of x is -1. Since -1 is not equal to -1 (which is the exception in the power rule), we can integrate it using the power rule. The integral of 1/x with respect to x is ln|x|.
For the second term, 39x^3/x, we can simplify it to 39x^2. We can then apply the power rule to integrate it as well. The integral of x^2 with respect to x is (1/3)x^3.
Now, combining both integrals, we have the indefinite integral of gc(x) = (8 + 39x^3) / x as ln|x| + (1/3)x^3 + C, where C is the constant of integration.
2) To find the indefinite integral of the function hc(u) = sin^2(1/8u), we can use a trigonometric identity. The identity states that sin^2(u) = (1/2)(1 - cos(2u)).
In this case, we have hc(u) = sin^2(1/8u). We can apply the trigonometric identity to simplify it to hc(u) = (1/2)(1 - cos(2(1/8u))). Simplifying further, we have hc(u) = (1/2)(1 - cos(1/4u)).
To integrate this function, we can use the power rule and apply a substitution. Let's substitute 1/4u as t. Then, dt = (1/4)du.
Now, our integral becomes (1/2)(1 - cos(t)) dt. This is a straightforward integral using the power rule. The integral of 1 with respect to t is t, and the integral of cos(t) with respect to t is sin(t).
Substituting t back as 1/4u, we have the indefinite integral of hc(u) = (1/2)(1 - cos(1/4u)) du as (1/2)(1 - cos(1/4u)) (1/4) + C, where C is the constant of integration.
3) To evaluate the integral ∫ (x(8 - 5x^2)) dx, we can use the distributive property of integration.
Expanding the expression, we have ∫ (8x - 5x^3) dx.
Applying the power rule for integration, we have ∫ (8x - 5x^3) dx = 4x^2 - (5/4)x^4 + C, where C is the constant of integration.
Therefore, the value of ∫ (x(8 - 5x^2)) dx is 4x^2 - (5/4)x^4 + C.