Find the integrals of the function: \(\frac{cos x - sin x}{1+sin 2x}\)
Solution and Explanation
\(\frac{cos x - sin x}{1+sin 2x}\) \(= \frac{cos x -sin x}{(sin^2 x + cos^2 x)+2sin x cos x }\) [sin2 x+cos2 x =1; sin 2x=2sin x cos x]
\(=\frac{cos x-sin x}{(sin x+cos x)^2}\)
Let \(sin x +cos x = t\)
\(∴ (cos x-sin x)dx = dt\)
⇒ ∫\(\frac{cos x - sin x}{1+sin 2x}\) = ∫\(\frac {cos x-sin x}{(sin x + cos x)^2}\) dx
\(= ∫\frac{dt}{t^2}\)
\(= ∫t^2dt\)
\(=-t^2+C\)
\(=-\frac{1}{t}+C\)
\(=\frac{-1}{sin x +cos x} +C\)
Top Questions on integral
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Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
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