
What is equal to? (C is an arbitrary constant)
[a] xsinx+C
[b] xcosx+C
[c] -xsinx+C
[d] -xcosx+C
Answer
504.3k+ views
Hint: Use the fact that and hence prove that the given integral is equal to . In the first integral use integration by parts taking x as first function and cosx as second function as per the ILATE rule. The result of the first function and second function can be related and hence find the value of the given integral.
Complete step-by-step answer:
Let
We know that (Linearity of integration).
Hence, we have
, where and
Evaluating
We know that . This is known as integration by parts rule. Here is called the first function and is known as the second function.
While solving questions using integration by parts, it is important to choose the first function and second function in such a way that the integral simplifies. A general order of preference for first function is given by ILATE rule
I = Inverse Trigonometric
L = Logarithmic
A = Algebraic
T = Trigonometric
E = Exponential.
Hence, according to the ILATE rule, we choose u(x) = x and v(x) = cosx.
We have and
Hence, we have
We have
Hence, we have
Adding on both sides, we get
But we know that
Hence, we have
So, the correct answer is “Option A”.
Note: [1] Verification: In case the integrands are simple, we should always verify the correctness of our solution. We can verify the correctness of our solution by checking that the derivative of our answer is equal to the integrand.
We have
Hence, we have
Hence our solution is verified to be correct.
Complete step-by-step answer:
Let
We know that
Hence, we have
Evaluating
We know that
While solving questions using integration by parts, it is important to choose the first function and second function in such a way that the integral simplifies. A general order of preference for first function is given by ILATE rule
I = Inverse Trigonometric
L = Logarithmic
A = Algebraic
T = Trigonometric
E = Exponential.
Hence, according to the ILATE rule, we choose u(x) = x and v(x) = cosx.
We have
Hence, we have
We have
Hence, we have
Adding
But we know that
Hence, we have
So, the correct answer is “Option A”.
Note: [1] Verification: In case the integrands are simple, we should always verify the correctness of our solution. We can verify the correctness of our solution by checking that the derivative of our answer is equal to the integrand.
We have
Hence, we have
Hence our solution is verified to be correct.
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