To antidifferentiate sin(3x), we can use the integration formula for the sine function:
∫sin(x) dx = -cos(x) + C
Since the argument of the sine function is 3x, we need to adjust the formula accordingly. We divide the argument by 3 to make it sin(3x):
∫sin(3x) dx = ∫(1/3) * 3sin(3x) dx
Now, we can apply the formula and adjust for the constant factor of 1/3:
= (1/3) * (-cos(3x)) + C
Therefore, the antidifferentiation of sin(3x) is (1/3) * (-cos(3x)) + C.
i understand upto sin 3x is written as -cos(3x+c) cna you explain the toher steps more in details
Certainly! Let's go through the steps in more detail.
1. Start with the integral: ∫sin(3x) dx.
2. We can rewrite sin(3x) as (1/3) * 3sin(3x) to match the form of the integration formula for sin(x).
3. Next, we use the formula for the integral of sin(x): ∫sin(x) dx = -cos(x) + C. Since the argument of the sine function is 3x, we replace x with 3x in the formula:
∫(1/3) * 3sin(3x) dx = (1/3) * -cos(3x) + C.
4. Finally, we simplify the expression by multiplying the constant factor (1/3) by -cos(3x):
= (1/3) * (-cos(3x)) + C.
So, the antidifferentiation (or integrating) of sin(3x) is (1/3) * (-cos(3x)) + C, where C is the constant of integration.
Certainly! Let's break down the steps in more detail:
Step 1:
We start with the integration formula ∫sin(x) dx = -cos(x) + C, which gives us the antiderivative of sin(x) (where C is the constant of integration).
Step 2:
To antidifferentiate sin(3x), we notice that the argument of the sine function is 3x instead of x. So, we want to transform sin(3x) into a form where we can use the integration formula. One way to do this is by introducing a constant factor of 3.
Step 3:
By multiplying and dividing by 3, we can rewrite sin(3x) as (1/3) * 3sin(3x). This step is not necessary, but it helps us match the form of the integration formula.
Step 4:
Now that we have (1/3) * 3sin(3x), we can apply the integration formula to ∫sin(3x) dx. Using the formula, we get:
∫sin(3x) dx = ∫(1/3) * 3sin(3x) dx = (1/3) * ∫3sin(3x) dx
Step 5:
Applying the formula ∫sin(x) dx = -cos(x) + C, we integrate ∫3sin(3x) dx. The constant factor of 3 remains outside the integral:
(1/3) * ∫3sin(3x) dx = (1/3) * (-cos(3x)) + C
Step 6:
Finally, simplify the expression by multiplying the constant factor (1/3) to the result of the integration and combine with the constant of integration C. This gives us the final solution:
(1/3) * (-cos(3x)) + C
So, the antidifferentiation of sin(3x) is (1/3) * (-cos(3x)) + C.
Sure! Let's go through the steps in more detail.
To antidifferentiate sin(3x), we start by recognizing that the derivative of -cos(3x) is sin(3x). This is because if we take the derivative of -cos(3x) with respect to x, we get (-1) * (-3sin(3x)), which simplifies to 3sin(3x).
Now, let's use this information to find the antidifferentiation of sin(3x). We can rewrite sin(3x) as (1/3) * 3sin(3x), which allows us to apply the linearity property of integrals.
∫sin(3x) dx = ∫(1/3) * 3sin(3x) dx
Using the linearity property, we can now integrate each term separately:
= (1/3) * ∫3sin(3x) dx
The integral of 3sin(3x) can be found by using the integration formula for the sine function:
∫sin(x) dx = -cos(x) + C
To adjust the formula for the argument 3x, we need to divide the argument by 3:
∫sin(3x) dx = ∫(1/3) * 3sin(3x) dx
Now, we can apply the formula and adjust for the constant factor of 1/3:
= (1/3) * (-cos(3x)) + C
Therefore, the antidifferentiation of sin(3x) is (1/3) * (-cos(3x)) + C.