evaluate the following indefinite integrals by substitution & check the result by differentiation.
∫(sin2x)^2 cos2xdx
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integrate (sin(2x))^2cos(2x)dx
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derivative (1/6)sin(2x)^3
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or, just look at the problem. Substitute
u = sin 2x
du = 2cos2x dx
and you have
∫u^2 du/2
If that gives you trouble, you got some catching up to do...
To evaluate the indefinite integral
∫ (sin^2x)^2 * cos(2x) dx
using the substitution method, follow these steps:
Step 1: Select the substitution variable
Let u = sin^2(x). This choice allows us to simplify the expression and make the integral more manageable.
Step 2: Compute the derivative of u with respect to x
Differentiating both sides of the equation with respect to x gives:
du/dx = d(sin^2(x))/dx
du/dx = 2sin(x) * cos(x)
Step 3: Rearrange the integral in terms of u
Substitute u = sin^2(x) and du = 2sin(x) * cos(x) dx into the integral expression:
∫ (sin^2(x))^2 * cos(2x) dx = ∫ u^2 * cos(2x) dx
To express the integral in terms of u instead of x, we need to replace dx with du/(2sin(x) * cos(x)). Recall that sin(2x) = 2sin(x) * cos(x). Therefore, cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x).
Step 4: Simplify the integral expression
∫ u^2 * (1 - 2sin^2(x)) * (1/(2sin(x) * cos(x))) du
Now we can simplify further:
∫ (u^2 * (1 - 2sin^2(x)))/(2sin(x) * cos(x)) du
Step 5: Integrate with respect to u
Using the substitution u = sin^2(x), the integral simplifies to:
∫ (u^2 - 2u^3)/(2sin(x) * cos(x)) du
Step 6: Evaluate the integral
∫ u^2 / (2sin(x) * cos(x)) du = ∫ u^2 * (1/sin(2x)) du
Using the trigonometric identity, sin(2x) = 2sin(x) * cos(x), we can further simplify the expression:
∫ u^2 / (2sin(x) * cos(x)) du = ∫ (u^2/2) * (1/(2sin(x) * cos(x))) du
∫ u^2 / (2sin(x) * cos(x)) du = ∫ (u^2/4) * (1/(sin(x) * cos(x))) du
Step 7: Integrate with respect to u
Using the substitution v = sin(x) * cos(x), dv = (1/2) * (cos^2(x) - sin^2(x)) dx = (1/2) * (1 - 2sin^2(x)) dx, we can rewrite the integral as:
∫ (u^2/4) * (1/v) dv
Now, the integral expression becomes:
∫ (u^2/4v) dv = (1/4) * ∫ (u^2/v) dv
Step 8: Integrate with respect to v
Integrating, we get:
(1/4) * ∫ (u^2/v) dv = (1/4) * (u^2 * ln|v|) + C
Step 9: Substitute back in terms of x
Since v = sin(x) * cos(x), we have:
(1/4) * (u^2 * ln|v|) + C = (1/4) * (sin^2(x) * ln|sin(x) * cos(x)|) + C
Thus, the indefinite integral ∫ (sin^2(x))^2 * cos(2x) dx is equal to:
(1/4) * (sin^2(x) * ln|sin(x) * cos(x)|) + C
To check the result by differentiation, compute the derivative of the obtained expression and verify if it matches the original integrand.