1.∫ csc (u-π/2) cot (u-π/2) du.
2.∫ sec^2(3x+2)ds
3.∫ √3-2s ds
4. ∫ 12(y^4+y4y^2+1)^2 (4^3+2y) dy use u= y^4+4y^2+1
Physics? Looks like elem calculus to me.
Its physics
1.
Use substitution x=u-&pi/2; dx=du
∫csc(u-π/2)cot(u-π/2)du
=∫csc(x)cot(x)dx
=∫cos(x)dx/sin²(x)
=∫d(sin(x))sin²(x)
=-1/sin(x)+C
1. To solve the integral ∫ csc(u-π/2) cot(u-π/2) du, we can use a trigonometric identity to simplify the expression:
csc(u-π/2) = 1/sin(u-π/2) = 1/cos(u)
cot(u-π/2) = 1/tan(u-π/2) = 1/cot(u)
Now, we can rewrite the integral as ∫ (1/cos(u)) * (1/cot(u)) du = ∫ (1/cos(u)) * (tan(u)) du
Next, let's use a substitution:
Let z = sin(u), so dz = cos(u) du
By rewriting the integral in terms of z, we have:
∫ (1/cos(u)) * (tan(u)) du = ∫ (1/z) dz
The integral of (1/z) with respect to z is ln|z| + C, where C is the constant of integration.
Finally, substituting z back in terms of u, we have:
∫ csc(u-π/2) cot(u-π/2) du = ln|sin(u)| + C
2. To find the integral of ∫ sec^2(3x+2) ds, we need to clarify what you mean by "ds". Are you referring to the differential of a variable other than x (such as s)? Please provide clarification so that I can assist you further.
3. The integral of ∫ √(3-2s) ds can be solved using a simple substitution:
Let u = 3-2s, so du = -2ds
Rearranging the equation, we have ds = -du/2
Substituting for ds in the integral, we get:
∫ √(3-2s) ds = ∫ √u (-du/2)
Simplifying, we have:
-1/2 ∫ √u du
Now, integrate √u with respect to u:
-1/2 * (2/3) * u^(3/2) + C = -(1/3)u^(3/2) + C
Finally, substitute u back in terms of s:
∫ √(3-2s) ds = -(1/3)(3-2s)^(3/2) + C
4. To calculate the integral ∫ 12(y^4+y4y^2+1)^2 (4^3+2y) dy using the substitution u = y^4+4y^2+1, we can follow these steps:
Differentiate both sides of the substitution: du = (4y^3 + 8y) dy
Rewrite the integral in terms of u: ∫ 12u^2 (4^3+2y) (1/(4y^3+8y)) du
Simplify: ∫ 12u^2 (64+2y) (1/(4y^3+8y)) du
Substitute 1/(4y^3+8y) with 1/(4y(y^2+2)): ∫ 12u^2 (64+2y) (1/(4y(y^2+2))) du
Cancel out common terms: ∫ 3u^2 (64+2y)/(y(y^2+2)) du
Substitute back u = y^4+4y^2+1: ∫ 3(y^4+4y^2+1)^2 (64+2y)/(y(y^2+2)) dy
Simplifying further might require expanding and factoring the expression.