Find the integrals of the function: \(cos^4 2x\)
Solution and Explanation
cos4 2x = (cos2 2x)2
\(=(\frac{1+cos4x}{2})^2\)
=\(\frac14\)[1+cos2 4x+2cos 4x]
\(= (\frac{1}{4}) [1+(\frac{1+cos8x}{2})+2cos4x]\)
\(= (\frac{1}{4}) [1+\frac{1}{2}+\frac{cos8x}{2}+2cos4x]\)
\(= (\frac{1}{4}) [ \frac{3}{2}+\frac{cos8x}{2}+2cos4x]\)
∴ ∫cos4 2x dx = ∫\((\frac{3}{8}+\frac{cos8x}{8}+\frac{cos4x}{2})dx\)
\(=\frac{3}{8}x+\frac{sin8x}{64}+\frac{sin4x}{8}+C\)
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C