Question

Проинтегрировать функциюxsinx/cos^3x

Answer 1

Слушай а ты точно в шк учишься?

Answer 2

Integration of x*sin(x)/cos^3(x)

To integrate the function x*sin(x)/cos^3(x), we can use the technique of integration by parts. Integration by parts is a method that allows us to integrate the product of two functions by differentiating one function and integrating the other.

Let's denote the function x*sin(x)/cos^3(x) as f(x). To integrate f(x), we can choose u = x and dv = sin(x)/cos^3(x) dx. By differentiating u and integrating dv, we can find du and v, respectively.

Differentiating u = x, we get du = dx. Integrating dv = sin(x)/cos^3(x) dx, we can rewrite it as dv = sin(x)*cos^(-3)(x) dx.

Now, we can apply the formula for integration by parts, which states:

∫ u dv = u*v - ∫ v du

Using this formula, we can integrate f(x) as follows:

∫ x*sin(x)/cos^3(x) dx = x * ∫ sin(x)*cos^(-3)(x) dx - ∫ (∫ sin(x)*cos^(-3)(x) dx) dx

Let's evaluate each integral step by step.

Step 1: Evaluating the first integral

The first integral, ∫ sin(x)*cos^(-3)(x) dx, can be evaluated using a substitution. Let's substitute u = cos(x), which gives us du = -sin(x) dx. Rearranging the equation, we have dx = -du/sin(x).

Substituting these values into the integral, we get:

∫ sin(x)*cos^(-3)(x) dx = ∫ -du/u^3

Integrating -du/u^3, we have:

∫ -du/u^3 = u^(-2)/2 + C1

Substituting u = cos(x) back into the equation, we get:

∫ sin(x)*cos^(-3)(x) dx = cos^(-2)(x)/2 + C1

Step 2: Evaluating the second integral

Now, let's evaluate the second integral, ∫ (∫ sin(x)*cos^(-3)(x) dx) dx.

Using the result from Step 1, we can rewrite the second integral as:

∫ (∫ sin(x)*cos^(-3)(x) dx) dx = ∫ (cos^(-2)(x)/2 + C1) dx

Expanding the integral, we get:

∫ (cos^(-2)(x)/2 + C1) dx = ∫ cos^(-2)(x)/2 dx + ∫ C1 dx

The integral of C1 dx is simply C1x.

To evaluate the integral of cos^(-2)(x)/2, we can use the trigonometric identity:

cos^(-2)(x) = sec^2(x)

Using this identity, the integral becomes:

∫ cos^(-2)(x)/2 dx = ∫ sec^2(x)/2 dx

The integral of sec^2(x) is tan(x), so we have:

∫ sec^2(x)/2 dx = (1/2)∫ sec^2(x) dx = (1/2)tan(x) + C2

Combining the results, we have:

∫ (∫ sin(x)*cos^(-3)(x) dx) dx = (1/2)tan(x) + C2 + C1x

Step 3: Final result

Finally, substituting the results from Step 1 and Step 2 into the original integral, we get:

∫ x*sin(x)/cos^3(x) dx = x * (cos^(-2)(x)/2 + C1) - (1/2)tan(x) - C2 - C1x

Simplifying the expression, we have:

∫ x*sin(x)/cos^3(x) dx = (x*cos^(-2)(x))/2 - (1/2)tan(x) - C1x - C2

Therefore, the integral of x*sin(x)/cos^3(x) is given by:

∫ x*sin(x)/cos^3(x) dx = (x*cos^(-2)(x))/2 - (1/2)tan(x) - C1x - C2