1. Evaluate the indefinite integral
([6x^2 + 12x^(3/2) +4x+9]/sqrt x)dx.
Answer = + C
2. Evaluate the indefinite integral
(12sin x+4tan x)dx.
Answer = + C
3. Evaluate the indefinite integral.
(x^7)e^(x^8)dx.
Answer = + C
Thank you for helping.
∫(6x^2 + 12x^(3/2) +4x+9]/√x)
= ∫ 6x^(3/2) + 12x + 4x^(1/2) + 9x^(-1/2) dx
= (12/5)x^(5/2) = 6x^2 + (8/3)x^(3/2) + 18x^(1/2) + c
∫ (12sinx + 4tanx) dx
= ∫( 12sinx + 4 sinx/cosx) dx
= -12cosx - 4ln(cosx) + c
∫ (x^7) e^(x^8) dx
= (1/8) e^(x^8) + c
To evaluate these indefinite integrals, we can use different integration techniques such as substitution, integration by parts, and trigonometric identities. Let's go through each question and explain how to find the integrals.
1. Evaluate the indefinite integral
([6x^2 + 12x^(3/2) +4x+9]/sqrt x)dx.
To integrate this expression, we can use substitution. Let's consider the substitution u = x^(1/2). Then, we can rewrite the integral as:
∫ [(6u^4 + 12u^3 + 4u^2 + 9)/u] du.
Now, we can simplify the expression:
∫ (6u^3 + 12u^2 + 4u + 9) du.
Integrating each term separately, we get:
(6/4)u^4 + (12/3)u^3 + (4/2)u^2 + 9u + C.
Substituting back u = x^(1/2), we have:
(3/2)x^2 + 4x^(3/2) + 2x + 9√(x) + C.
So, the indefinite integral is (3/2)x^2 + 4x^(3/2) + 2x + 9√(x) + C.
2. Evaluate the indefinite integral
(12sin x+4tan x)dx.
In this case, we can use trigonometric identities to simplify and integrate the expression. We know that the integral of sin(x)dx is -cos(x) + C and the integral of tan(x)dx is ln|sec(x)| + C.
Therefore, the indefinite integral becomes:
∫ (12sin x + 4tan x)dx = -12cos(x) + 4ln|sec(x)| + C.
So, the indefinite integral is -12cos(x) + 4ln|sec(x)| + C.
3. Evaluate the indefinite integral.
(x^7)e^(x^8)dx.
In this case, we can use substitution. Let's consider the substitution u = x^8. Then, we can rewrite the integral as:
∫ (1/8)(u^(1/8))e^u du.
Now, we can simplify the expression:
(1/8) ∫ u^(1/8)e^u du.
Integrating this expression requires using special functions known as the exponential integral or gamma function, which are beyond the scope of this explanation. So, the final answer would be:
(1/8) * Ei(u^(1/8)) + C,
where Ei represents the exponential integral function. Substituting back u = x^8, we have:
(1/8) * Ei(x^(1/8)) + C.
So, the indefinite integral is (1/8) * Ei(x^(1/8)) + C.
Note that in some cases, the integrals may not have an elementary function as an antiderivative, and we need to resort to special functions or numerical methods to evaluate them.