y the integration by parts, we haveZ π−πf (x) cos(nx)dx
Question
y the integration by parts, we haveZ π−πf (x) cos(nx)dx
Solution
To solve the integral ∫(π to -π) f(x) cos(nx) dx using integration by parts, we can follow these steps:
Step 1: Choose u and dv Let u = f(x) and dv = cos(nx) dx.
Step 2: Find du and v Differentiate u to find du, and integrate dv to find v. du = f'(x) dx v = (1/n) sin(nx)
Step 3: Apply the integration by parts formula The integration by parts formula states that ∫u dv = uv - ∫v du. Using this formula, we can rewrite the integral as: ∫(π to -π) f(x) cos(nx) dx = [f(x) * (1/n) sin(nx)] - ∫(π to -π) (1/n) sin(nx) f'(x) dx
Step 4: Simplify the integral The integral on the right-hand side can be simplified further by using integration by parts again. Repeat steps 1 to 3 for this new integral until you reach a point where the integral can be easily evaluated.
Step 5: Evaluate the integral Once you have simplified the integral, evaluate it using the given limits of integration (π to -π) and any other information provided about the function f(x).
Note: The specific steps and calculations may vary depending on the function f(x) and the value of n.
Similar Questions
∫ e^x cos(x) dx using integration by parts
sinn x dx = − 1n sinn−1 x · cos x + n − 1n∫sinn−2 x dx. (6.4)If we take dv = sin x dx, then we have v = − cos x and we may integrate by parts withu = sinn−1 x, du = (n − 1) sinn−2 x · cos x.Using the fact that sin2 x + cos2 x = 1, one may thus conclude that∫sinn x dx = − sinn−1 x · cos x + (n − 1)∫sinn−2 x · cos2 x dx= − sinn−1 x · cos x + (n − 1)∫sinn−2 x · (1 − sin2 x) dx= − sinn−1 x · cos x + (n − 1)∫sinn−2 x dx + (1 − n)∫sinn x dx.Here, the rightmost integral coincides with the original integral on the left. Once we nowrearrange terms, we end up with n copies of the integral and equation (6.4) follows. Example 6.11 We use a reduction formula to compute the integral I3 in the case thatIn =∫xne2x dx.If we take u = xn and dv = e2x dx, then du = nxn−1 dx and v = 12 e2x, so one hasIn = 12 xne2x − n2∫xn−1e2x dx = 12 xne2x − n2 · In−1. (6.5)We now apply the last formula repeatedly to determine I3. According to the formula,I3 = 12 x3e2x − 32 · I2 = 12 x3e2x − 32 ·[12 x2e2x − I1]= 12 x3e2x − 32 ·[12 x2e2x − 12 xe2x + 12 · I0]= 12 x3e2x − 34 x2e2x + 34 xe2x − 34∫e2x dx= 12 x3e2x − 34 x2e2x + 34 xe2x − 38 e2x + C.
[1 - cos(nπ/2) - sin(nπ/2)]^2, Z transform of the question
1) Integrate the following functions: f(x)=cos(3x)
The area under the curve y=|cosx−sinx|, 0≤x≤π2, and above x-axis is :
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