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.