
How do you integrate \[\int{x\sin x}\] by integration by parts method?
Answer
447.6k+ views
Hint: From the question given, we have been asked to integrate \[\int{x\sin x}\] by integration by parts method. To solve the given question by using the integration by parts formula, first of all, we have to know the formula for integration by parts. If we are studying integration, then we should have learnt the formula for integration by parts and how to use it. The formula for integration by parts is given as \[\int{u\dfrac{dv}{dx}dx=uv-\int{v\dfrac{du}{dx}}}\]
Complete step by step answer:
From the given question, we have been given that \[\int{x\sin x}\]
Let \[u=x\]
Then, derivative for it is shown below \[\dfrac{du}{dx}=1\]
And
\[\dfrac{dv}{dx}=\sin x\]
\[\Rightarrow v=-\cos x\]
Now, we have got all the values which we need to substitute in the integration by parts formula.
Therefore substitute the values we got in the integration by parts formula to solve the given question.
By substituting all the values we got above in the integration by parts formula, we get
\[\int{u\dfrac{dv}{dx}dx=uv-\int{v\dfrac{du}{dx}}}\]
\[\Rightarrow \int{x\sin xdx=\left( x \right)\left( -\cos x \right)-\int{\left( -\cos x \right)\left( 1 \right)dx}}\]
Now, simplify further to get the exact answer.
By simplifying furthermore, we get
\[\Rightarrow \int{x\sin xdx=-x\cos x+\int{\cos xdx}}\]
\[\Rightarrow \int{x\sin xdx=-x\cos x+\sin x+c}\]
Hence, the given question is solved by using the integration by parts formula.
Note: We should be well known about the integration by parts formula. Also, we should be well known about the application of integration by parts formula. Also, we should be very careful while doing the calculation part of the problem. Also, we should be very careful while applying the integration by parts formula. Similarly we have many more integration formulae like $\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1},n\ne -1}$ and many more.
Complete step by step answer:
From the given question, we have been given that \[\int{x\sin x}\]
Let \[u=x\]
Then, derivative for it is shown below \[\dfrac{du}{dx}=1\]
And
\[\dfrac{dv}{dx}=\sin x\]
\[\Rightarrow v=-\cos x\]
Now, we have got all the values which we need to substitute in the integration by parts formula.
Therefore substitute the values we got in the integration by parts formula to solve the given question.
By substituting all the values we got above in the integration by parts formula, we get
\[\int{u\dfrac{dv}{dx}dx=uv-\int{v\dfrac{du}{dx}}}\]
\[\Rightarrow \int{x\sin xdx=\left( x \right)\left( -\cos x \right)-\int{\left( -\cos x \right)\left( 1 \right)dx}}\]
Now, simplify further to get the exact answer.
By simplifying furthermore, we get
\[\Rightarrow \int{x\sin xdx=-x\cos x+\int{\cos xdx}}\]
\[\Rightarrow \int{x\sin xdx=-x\cos x+\sin x+c}\]
Hence, the given question is solved by using the integration by parts formula.
Note: We should be well known about the integration by parts formula. Also, we should be well known about the application of integration by parts formula. Also, we should be very careful while doing the calculation part of the problem. Also, we should be very careful while applying the integration by parts formula. Similarly we have many more integration formulae like $\int{{{x}^{n}}dx=\dfrac{{{x}^{n+1}}}{n+1},n\ne -1}$ and many more.
Recently Updated Pages
Master Class 12 Economics: Engaging Questions & Answers for Success

Master Class 12 Maths: Engaging Questions & Answers for Success

Master Class 12 Biology: Engaging Questions & Answers for Success

Master Class 12 Physics: Engaging Questions & Answers for Success

Master Class 12 Business Studies: Engaging Questions & Answers for Success

Master Class 12 English: Engaging Questions & Answers for Success

Trending doubts
Who is Mukesh What is his dream Why does it look like class 12 english CBSE

Who was RajKumar Shukla Why did he come to Lucknow class 12 english CBSE

The word Maasai is derived from the word Maa Maasai class 12 social science CBSE

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

Why is the cell called the structural and functional class 12 biology CBSE

Which country did Danny Casey play for class 12 english CBSE
