I'm really having trouble with this current topic that we're learning. Any explanations are greatly appreciated.

Evaluate the integrals:

1.) ∫ (2-2cos^2x) dx

2.) ∫ cot3x dx

3.) ∫ ((e^(sqrt x) / (sqrt x) dx))

1.) ∫ (2-2cos^2x) dx

= ∫ (2-2cos^2x) dx
= ∫ (1-(2cos^2x-1)) dx
= ∫ (1-cos 2x) dx
= x - (1/2)sin 2x + c

2.
∫ cot3x dx
= ∫ cos3x/sin 3x dx
= (1/3) ln(sin 3x) + c

3.
∫ (e^√x / √x dx
= ∫ e^(x^(1/2)) (e^(-1/2)) dx
= 2 e^(x^(1/2)) + c or 2 e^√x + c

Sure! I'd be happy to help you with these integrals.

1.) ∫ (2-2cos^2x) dx

To evaluate this integral, we need to expand the expression 2-2cos^2x.

Recall the identity: cos^2x = (1 + cos2x)/2.

Using this identity, we can simplify the expression:

2 - 2cos^2x = 2 - 2(1 + cos2x)/2
= 2 - (2 + 2cos2x)/2
= 2 - 1 - cos2x
= 1 - cos2x.

Now, we can rewrite our integral as:

∫ (2-2cos^2x) dx = ∫ (1 - cos2x) dx

Integrating term by term, we have:

∫ (1 - cos2x) dx = ∫ 1 dx - ∫ cos2x dx
= x - (1/2)sin2x + C.

So the solution to the integral is x - (1/2)sin2x + C, where C is the constant of integration.

2.) ∫ cot3x dx

To evaluate this integral, we can use the trigonometric identity cotx = 1/tanx.

Rewriting the integral using this identity, we get:

∫ cot3x dx = ∫ (1/tan3x) dx.

Now, using the inverse property of integration, we can rewrite the integral as:

∫ (1/tan3x) dx = ∫ (tan^(-1)(3x)) dx.

Using the derivative property of the inverse tangent function, we have:

∫ (tan^(-1)(3x)) dx = (1/3)∫ (tan^(-1)(u)) du,
where u = 3x.

The integral of tan^(-1)(u) can be evaluated using integration by parts or by looking it up in a table of integrals. Using the latter approach, we find:

∫ (tan^(-1)(u)) du = u * tan^(-1)(u) - ∫ (u/(1+u^2)) du
= u * tan^(-1)(u) - (1/2)ln|1+u^2| + C.

Substituting u = 3x back in, we get:

∫ cot3x dx = (1/3)(3x * tan^(-1)(3x) - (1/2)ln|1+(3x)^2|) + C
= x * tan^(-1)(3x) - (1/2)ln|1+9x^2| + C.

So the solution to the integral is x * tan^(-1)(3x) - (1/2)ln|1+9x^2| + C, where C is the constant of integration.

3.) ∫ ((e^(sqrt x) / (sqrt x)) dx

To evaluate this integral, we can use the substitution method.

Let u = √x, so du = (1/2)(x^(-1/2))dx.

Substituting these into the integral, we get:

∫ ((e^(√x) / √x) dx = ∫ (2e^u) du.

Now, we can integrate the right-hand side of the equation:

∫ (2e^u) du = 2∫ e^u du
= 2e^u + C.

Substituting u = √x back in, we have:

∫ ((e^(√x) / √x) dx = 2e^(√x) + C.

So the solution to the integral is 2e^(√x) + C, where C is the constant of integration.

I hope these explanations help you understand how to evaluate these integrals! If you have any further questions, feel free to ask.