∫ (dx/(sinx + cosx)).dx = ?
∫ (dx/(sinx + cosx)).dx = ?
Solution and Explanation
To solve the integral ∫ (dx / (sin(x) + cos(x))), we can use the following steps:
1. Rewrite the denominator:
We can rewrite sin(x) + cos(x) as √2 * (1/√2 * sin(x) + 1/√2 * cos(x)).
Since cos(π/4) = 1/√2 and sin(π/4) = 1/√2, we can write:
sin(x) + cos(x) = √2 * (cos(π/4) * sin(x) + sin(π/4) * cos(x)).
Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we get:
sin(x) + cos(x) = √2 * sin(x + π/4).
2. Substitute into the integral:
∫ (dx / (sin(x) + cos(x))) = ∫ (dx / (√2 * sin(x + π/4))).
3. Simplify and integrate:
∫ (dx / (√2 * sin(x + π/4))) = (1/√2) * ∫ csc(x + π/4) dx.
We know that ∫ csc(u) du = ln|csc(u) - cot(u)| + C.
Therefore, (1/√2) * ∫ csc(x + π/4) dx = (1/√2) * ln|csc(x + π/4) - cot(x + π/4)| + C.
Thus, the solution is:
∫ (dx / (sin(x) + cos(x))) = (1/√2) * ln|csc(x + π/4) - cot(x + π/4)| + 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