1.∫ x³(x+1)dx
2.∫ (1-cot²x)dx
3.∫ cosx(tan x + sec x) dx
4.∫ dx/√5x+8 using u= 5x+8 and using u= √5x+5
Hey, I told you how to do these yesterday except number 4 which is incomprehensible without adding parentheses and telling me what is u and what is du. Come on, we are volunteers here.
integral x^4 dx = (1/5) x^5 + c
integral 1 dx = x + c
sum = (1/5) x^5 + x + c
1. To solve the integral ∫ x³(x+1)dx, you can expand the expression inside the integral and then integrate each term separately.
Expanding the expression: x³(x+1) = x⁴ + x³
Now, you can integrate each term separately:
∫ x⁴ dx = (1/5) x⁵ + C (integration rule: ∫ xⁿ dx = (1/(n+1)) x^(n+1) + C)
∫ x³ dx = (1/4) x⁴ + C
Adding the integrated terms:
∫ x³(x+1)dx = (1/5) x⁵ + (1/4) x⁴ + C
2. To solve the integral ∫ (1 - cot²x)dx, you can rewrite the expression using trigonometric identities.
Using the identity: cot²x = csc²x - 1
∫ (1 - cot²x)dx = ∫ (1 - (csc²x - 1))dx = ∫ (2 - csc²x)dx
Now, rewrite csc²x as 1/sin²x:
∫ (2 - csc²x)dx = ∫ (2 - 1/sin²x)dx
To simplify further, you can rewrite sin²x as 1 - cos²x:
∫ (2 - 1/(1 - cos²x))dx
Now, substitute u = cos(x):
du = -sin(x) dx
Replacing the variables:
∫ (2 - 1/(1 - u²))(-1/sin(x))dx
Simplifying:
∫ (1 - 1/(1 - u²))/sin(x) dx
The integral now becomes:
∫ (1 - sec²(x))/(sin(x)) dx
Using the identity: sec²(x) = 1 + tan²(x)
∫ (1 - (1 + tan²(x)))/(sin(x)) dx
∫ (-tan²(x))/(sin(x)) dx
Finally, you can substitute tan(x) = u:
∫ -u² du
Integrating:
- (1/3) u³ + C
Replacing u with tan(x):
- (1/3) tan³(x) + C
3. To solve the integral ∫ cos(x)(tan(x) + sec(x)) dx, you can use integration by substitution.
Let u = tan(x), then du = sec²(x) dx
The integral becomes:
∫ cos(x)((u) + sec(x)) du
Expanding the expression:
∫ (u cos(x) + sec(x) cos(x)) du
Distributing:
∫ (u cos(x)) du + ∫ (sec(x) cos(x)) du
The first integral:
∫ (u cos(x)) du can be integrated using the power rule:
(u sin(x)) - ∫ (sin(x)) du = u sin(x) - (-cos(x)) + C₁ = u sin(x) + cos(x) + C₁
The second integral:
∫ (sec(x) cos(x)) du can be solved using u-substitution. Let v = sec(x), then dv = sec(x) tan(x) dx.
The integral becomes:
∫ v dv
(v²/2) + C₂
Combining the integrated terms:
(u sin(x) + cos(x) + C₁) + ((sec(x))²/2) + C₂
Simplifying:
u sin(x) + cos(x) + (sec²(x)/2) + C
Replacing u with tan(x):
tan(x) sin(x) + cos(x) + (sec²(x)/2) + C
4. To solve the integral ∫ dx/√(5x+8), you can use substitution.
Option 1: Substituting u = 5x + 8
Find du/dx:
du/dx = 5
Rearrange to find dx:
dx = (1/5) du
Substitute dx and u into the integral:
∫ (1/√5) (1/u) du
Simplifying:
(1/√5) ∫ du/u
Integrating:
(1/√5) ln|u| + C
Replacing u with 5x + 8:
(1/√5) ln|5x + 8| + C
Option 2: Substituting u = √(5x + 8)
Find du/dx:
du/dx = (5/2) (1/√(5x+8))
Rearrange to find dx:
dx = (2/5) √(5x+8) du
Substitute dx and u into the integral:
∫ (1/√5) (2/5) √(5x+8) du
Simplifying:
(2/5√5) ∫ √u du
Integrating:
(2/5√5) (2/3) u^(3/2) + C
Simplifying further and replacing u with √(5x+8):
(4/15√5) (5x+8)^(3/2) + C