Find the integrals of the function: \(\frac{1}{cos(x-a)cos(x-b)}\)
Solution and Explanation
\(\frac{1}{cos(x-a)cos(x-b)} = \frac{1}{sin(a-b)}[\frac{sin(a-b)}{cos(x-a)cos(x-b)}]\)
\(=\frac{1}{sin(a-b)}[\frac{sin[(x-b)-(x-a)]}{cos(x-a)cos(x-b)}]\)
\(=\frac{1}{sin(a-b)}\frac{[sin(x-b)cos(x-a)-cos(x-b)sin(x-a)]}{cos(x-a)cos(x-b)}\)
\(=\frac{1}{sin(a-b)}[tan(x-b)-tan(x-a)]\)
\(⇒ ∫\frac{1}{cos(x-a)cos(x-b)}dx = \frac{1}{sin(a-b)} ∫[tan(x-b)-tan(x-a)]dx\)
\(=\frac{1}{sin(a-b)}[-log|cos(x-b)|+log|cos(x-a)|]\)
\(=\frac{1}{sin(a-b)}[log|\frac{cos(x-a)}{cos(x-b)}|]+C\)
Top Questions on integral
- The value of the integral \( \int_0^1 x^2 \, dx \) is:
Let \( f : (0, \infty) \to \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0 \), } \[ \int_0^a f(x) \, dx = f(a), \quad f(1) = 1, \quad f(16) = \frac{1}{8}, \quad \text{then } 16 - f^{-1}\left( \frac{1}{16} \right) \text{ is equal to:}\]
- Let $ f(x) $ be a positive function and $I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) dx$ and $I_2 = \int_{-1}^2 f\left(x(1-x)\right) dx.$ Then the value of $\frac{I_2}{I_1}$ is equal to ____
- Evaluate the integral: \[ \int \frac{x^2 + 2x}{\sqrt{x^2 + 1}} \, dx \]
- Evaluate the integral: \[ \int \sqrt{x^2 + 3x} \, dx \]
Questions Asked in CBSE CLASS XII exam
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
(i) target is hit.
(ii) at least one shot misses the target.
- CBSE CLASS XII - 2026
- Probability
- Identify and explain the various other forms of this disorder.
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- Psychological Disorders
- If \[ P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \quad \text{and} \quad Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 1 & -1 & 5 \end{bmatrix} \] find \( QP \) and hence solve the following system of equations using matrix method:
\[ x - y = 3,\quad 2x + 3y + 4z = 13,\quad y + 2z = 7 \] - Following reaction takes place in one step :
\( 2A + B \rightarrow 2C \)
How will the rate of above reaction change if the volume of the reaction vessel is decreased to one third of its original volume? Will there be any change in the order of reaction with the reduced volume?
- CBSE CLASS XII - 2026
- Chemical Kinetics
- In a Young's double-slit experiment, two waves each of intensity I superpose each other and produce an interference pattern. Prove that the resultant intensities at maxima and minima are 4I and zero respectively.
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- Wave Optics
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