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
1. is integral x^4 dx + integral x dx
2. same deal, split into sum of two integrals
integral of cot^2 x = (-1/x)cot x - x
3. yet again
cos*sin/cos = cos :)
cos * 1/cos = 1 :)
4. I think you have a typo and I do not know what is u and what is du or actually what is in the original denominator
do you mean
dx / [(5x)^.5 + 8 ]
or what
1. To solve the integral of ∫x³(x+1)dx, we can simplify by expanding the expression:
= ∫(x^4 + x³)dx
Then, we can apply the power rule of integration, which states that:
∫x^n dx = (x^(n+1))/(n+1) + C
Using this rule, let's integrate each term separately:
= ∫x^4 dx + ∫x³ dx
= (x^5)/5 + (x^4)/4 + C
Therefore, the solution to ∫x³(x+1)dx is (x^5)/5 + (x^4)/4 + C, where C represents the constant of integration.
2. To solve the integral of ∫(1 - cot²x)dx, we can rewrite it using the trigonometric identity:
cot²x + 1 = csc²x
By substituting this identity into the integral, we get:
= ∫(csc²x - cot²x)dx
Now, we can use the known integral of csc²x and cot²x:
∫csc²x dx = -cotx + C
∫cot²x dx = cotx - x + C
Applying these formulas to our original integral, we get:
= -cotx + cotx - x + C
= -x + C
Thus, the solution to ∫(1 - cot²x)dx is -x + C, where C represents the constant of integration.
3. To solve the integral of ∫cosx(tanx + secx)dx, we can apply the distributive property of multiplication and rewrite it as:
∫(cosx * tanx + cosx * secx)dx
We then integrate each term separately:
∫cosx * tanx dx + ∫cosx * secx dx
We know that the derivative of tanx is sec²x and the derivative of secx is secx * tanx. Using this knowledge, we can rewrite the integrals:
= ∫sinx dx + ∫secx * tanx dx
The integral of sinx is -cosx, and the integral of secx * tanx can be found by substituting u = secx:
du = secx * tanx dx
Substituting back into the integral, we get:
= -cosx + ∫du
The integral of du is simply u:
= -cosx + u + C
Reintroducing the original substitution, we have:
= -cosx + secx + C
Therefore, the solution to ∫cosx(tanx + secx)dx is -cosx + secx + C, where C represents the constant of integration.
4. To solve the integral of ∫dx/√(5x+8), let's start by using the substitution u = 5x + 8:
u = 5x + 8
du/dx = 5
dx = du/5
Substituting into the integral, we have:
= ∫(du/5) / √(u)
Simplifying, we get:
= (1/5) ∫1/√(u) du
The integral of 1/√(u) can be found using the power rule, where the integral of u^n is (2√(u))/(2n+1). In this case, n = -1/2:
= (1/5) * (2√(u))/(2*(-1/2)+1) + C
Simplifying further, we get:
= (1/5) * (2√(u))/(1) + C
= (2/5) * √(u) + C
Remembering that we substituted u = 5x + 8, we can rewrite the answer in terms of x:
= (2/5) * √(5x + 8) + C
Therefore, the solution to ∫dx/√(5x+8) using the substitution u = 5x + 8 is (2/5) * √(5x + 8) + C, where C represents the constant of integration.
Similarly, if we were to use the substitution u = √(5x + 5), we would follow a similar procedure, but with a different substitution and integral result.