Question

A) int (b=pi/3, a=pi/4) xdx/cos^2x dx Б) int (b=4, a=0) 4x+3/sqrt(2x+1)+5 dx

Answer 1

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Answer 2

Let's solve both of these definite integrals one by one:

A) ∫[b=π/3, a=π/4] xdx/cos^2x dx

To solve this integral, we first need to find the antiderivative of x/cos^2(x) with respect to x. Let's do that:

∫ x/cos^2(x) dx

We can rewrite cos^2(x) as (1 - sin^2(x)) using the Pythagorean identity, and then use a substitution to make the integration easier.

Let's use u = sin(x), then du = cos(x) dx:

∫ x/(1 - sin^2(x)) dx ∫ x/(1 - u^2) du Now, we need to convert x in terms of u. We know that u = sin(x), so x = arcsin(u):

∫ arcsin(u)/(1 - u^2) du

Now, let's solve this integral:

Let I be the integral we need to find. Then, I = ∫ arcsin(u)/(1 - u^2) du

Now, let's use integration by parts to solve this integral:

Let dv = du, then v = u, and du = 1 du:

I = u * u - ∫ u * 1 du I = u^2 - ∫ u du I = u^2 - u^2/2 + C I = u^2/2 + C

Now, we need to convert u back to x:

I = (sin^2(x))/2 + C

Finally, we can evaluate the definite integral:

∫[b=π/3, a=π/4] xdx/cos^2x dx = [(sin^2(π/3))/2 - (sin^2(π/4))/2] = [(3/4)/2 - (1/2)/2] = (3/8) - (1/4) = 1/8

B) ∫[b=4, a=0] (4x+3)/√(2x+1)+5 dx

To solve this integral, we can use a substitution to make the integration easier.

Let's use u = 2x + 1, then du = 2 dx:

∫ (4x+3)/√(2x+1)+5 dx ∫ (4x+3)/√u + 5 dx ∫ (2/u + 3)/√u du Now, we can rewrite the integral in terms of u:

∫ (2/u + 3)/√u du

Let's solve this integral:

Let I be the integral we need to find. Then, I = ∫ (2/u + 3)/√u du

Split the fraction into two separate integrals:

I = ∫ (2/u)/√u du + ∫ (3)/√u du

Now, let's solve each integral:

1. ∫ (2/u)/√u du:

Let's use a substitution: v = √u, then dv = (1/2√u) du:

∫ (2/u)/√u du = ∫ 2/v dv = 2 * ∫ dv = 2v + C = 2√u + C

2. ∫ (3)/√u du:

Let's use another substitution: w = √u, then dw = (1/2√u) du:

∫ (3)/√u du = ∫ 3/w dw = 3 * ∫ dw = 3w + C = 3√u + C

Now, let's integrate the original integral:

I = 2√u + 3√u + C I = 5√u + C

Finally, we need to convert u back to x:

I = 5√(2x + 1) + C

Now, we can evaluate the definite integral:

∫[b=4, a=0] (4x+3)/√(2x+1)+5 dx = [5√(24 + 1) - 5√(20 + 1)] = [5√9 - 5√1] = [5 * 3 - 5 * 1] = 15 - 5 = 10

So, the value of the definite integral is 10.