
How do you calculate the antiderivative of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\]?
Answer
445.5k+ views
Hint: We know that antiderivative means integration. We need to find the integration of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\]. Here we have an indefinite integral. In the numerator we have sine double angle, we know the sine double angle formula that is \[\sin (2x) = 2.\sin x.\cos x\]. We substitute this in the given problem and then we integrate with respect to ‘x’.
Complete step-by-step solution:
Given \[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} \].
We know \[\sin (2x) = 2.\sin x.\cos x\].
The term inside the integral symbol is called the integrand.
Then the integrand becomes
\[\dfrac{{\sin (2x)}}{{\cos (x)}} = \dfrac{{2.\sin x.\cos x}}{{\cos x}}\]
Cancelling the cosine function we have,
\[\dfrac{{\sin (2x)}}{{\cos (x)}} = 2.\sin x.\]
Now applying the integration we have
\[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} = \int {2.\sin x} .dx\]
\[ = \int {2.\sin x} .dx\]
Taking constant term outside the integral we have,
\[ = 2\int {\sin x} .dx\]
Integrating we have,
\[ = - 2\cos x + c\]
Thus we have
The antiderivative of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\] is \[ - 2\cos x + c\]. Where ‘c’ is the integration constant.
Note: In the given above problem we have an indefinite integral, that is no upper and lower limit. Hence we add the integration constant ‘c’ after integrating. In a definite integral we will have an upper and lower limit, we don’t need to add integration constant in the case of definite integral. We have different integration rule:
The power rule: If we have a variable ‘x’ raised to a power ‘n’ then the integration is given by \[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \].
The constant coefficient rule: if we have an indefinite integral of \[K.f(x)\], where f(x) is some function and ‘K’ represent a constant then the integration is equal to the indefinite integral of f(x) multiplied by ‘K’. That is \[\int {K.f(x)dx = c\int {f(x)dx} } \].
The sum rule: if we have to integrate functions that are the sum of several terms, then we need to integrate each term in the sum separately. That is
\[\int {\left( {f(x) + g(x)} \right)dx = \int {f(x)dx} } + \int {g(x)dx} \]
For the difference rule we have to integrate each term in the integrand separately.
Complete step-by-step solution:
Given \[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} \].
We know \[\sin (2x) = 2.\sin x.\cos x\].
The term inside the integral symbol is called the integrand.
Then the integrand becomes
\[\dfrac{{\sin (2x)}}{{\cos (x)}} = \dfrac{{2.\sin x.\cos x}}{{\cos x}}\]
Cancelling the cosine function we have,
\[\dfrac{{\sin (2x)}}{{\cos (x)}} = 2.\sin x.\]
Now applying the integration we have
\[\int {\dfrac{{\sin (2x)}}{{\cos (x)}}dx} = \int {2.\sin x} .dx\]
\[ = \int {2.\sin x} .dx\]
Taking constant term outside the integral we have,
\[ = 2\int {\sin x} .dx\]
Integrating we have,
\[ = - 2\cos x + c\]
Thus we have
The antiderivative of \[\dfrac{{\sin (2x)}}{{\cos (x)}}dx\] is \[ - 2\cos x + c\]. Where ‘c’ is the integration constant.
Note: In the given above problem we have an indefinite integral, that is no upper and lower limit. Hence we add the integration constant ‘c’ after integrating. In a definite integral we will have an upper and lower limit, we don’t need to add integration constant in the case of definite integral. We have different integration rule:
The power rule: If we have a variable ‘x’ raised to a power ‘n’ then the integration is given by \[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \].
The constant coefficient rule: if we have an indefinite integral of \[K.f(x)\], where f(x) is some function and ‘K’ represent a constant then the integration is equal to the indefinite integral of f(x) multiplied by ‘K’. That is \[\int {K.f(x)dx = c\int {f(x)dx} } \].
The sum rule: if we have to integrate functions that are the sum of several terms, then we need to integrate each term in the sum separately. That is
\[\int {\left( {f(x) + g(x)} \right)dx = \int {f(x)dx} } + \int {g(x)dx} \]
For the difference rule we have to integrate each term in the integrand separately.
Recently Updated Pages
Master Class 11 Accountancy: Engaging Questions & Answers for Success

Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Trending doubts
Which are the Top 10 Largest Countries of the World?

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

What is a transformer Explain the principle construction class 12 physics CBSE

Draw a labelled sketch of the human eye class 12 physics CBSE

What are the major means of transport Explain each class 12 social science CBSE

What is the Full Form of PVC, PET, HDPE, LDPE, PP and PS ?
