Find the indefinite integral \int x\sin x^{2}dx

krypsojx 2021-11-23 Answered
Find the indefinite integral \(\displaystyle\int{x}{{\sin{{x}}}^{{{2}}}{\left.{d}{x}\right.}}\)

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Expert Answer

Phisecome
Answered 2021-11-24 Author has 393 answers
Step 1
Given: The integral \(\displaystyle\int{x}{{\sin{{x}}}^{{{2}}}{\left.{d}{x}\right.}}\),
To evaluate: The given indefinite integral.
Step 2
Explanation:
Let \(\displaystyle{I}=\int{x}{{\sin{{x}}}^{{{2}}}{\left.{d}{x}\right.}}\)
Substituting
\(\displaystyle{x}^{{{2}}}={t}\)
\(\displaystyle\Rightarrow{2}{x}{\left.{d}{x}\right.}={\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow{x}{\left.{d}{x}\right.}={\frac{{{\left.{d}{t}\right.}}}{{{2}}}}\)
Hence
\(\displaystyle{I}=\int{\frac{{{\sin{{t}}}{\left.{d}{t}\right.}}}{{{2}}}}\)
\(\displaystyle\Rightarrow{I}={\frac{{{1}}}{{{2}}}}\int{\sin{{t}}}{\left.{d}{t}\right.}\)
\(\displaystyle\Rightarrow{I}=-{\frac{{{1}}}{{{2}}}}{\cos{{t}}}+{C}\ \ \ {\left[\because\int{\sin{{x}}}{\left.{d}{x}\right.}=-{\cos{{x}}}\right]}\)
Now substituting \(\displaystyle{t}={x}^{{{2}}}\),
\(\displaystyle{I}=-{\frac{{{\cos{{x}}}^{{{2}}}}}{{{2}}}}+{C}\)
Where C is arbitrary constant known as constant of integration.
Answer: \(\displaystyle\int{x}{{\sin{{x}}}^{{{2}}}{\left.{d}{x}\right.}}=-{\frac{{{\cos{{x}}}^{{{2}}}}}{{{2}}}}+{C}\), where C is constant of integration.
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Befory
Answered 2021-11-25 Author has 178 answers
Step 1: Use Integration by Substitution.
Let \(\displaystyle{u}={x}^{{{2}}},{d}{u}={2}{x}{\left.{d}{x}\right.},\ {t}{h}{e}{n}\ {x}{\left.{d}{x}\right.}={\frac{{{1}}}{{{2}}}}{d}{u}\)
Step 2: Using u and du above, rewrite \(\displaystyle\int{x}{\sin{{\left({x}^{{{2}}}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle\int{\frac{{{\sin{{u}}}}}{{{2}}}}{d}{u}\)
Step 3: Use Constant Factor Rule: \(\displaystyle\int{c}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}={c}\int{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\frac{{{1}}}{{{2}}}}\int{\sin{{u}}}{d}{u}\)
Step 4: Use Trigonometric Integration: the integral of \(\displaystyle{\sin{{u}}}\ {i}{s}\ -{\cos{{u}}}\).
\(\displaystyle-{\frac{{{\cos{{u}}}}}{{{2}}}}\)
Step 5: Substitute \(\displaystyle{u}={x}^{{{2}}}\) back into the original integral.
\(\displaystyle-{\frac{{{\cos{{\left({x}^{{{2}}}\right)}}}}}{{{2}}}}\)
Step 6: Add constant.
\(\displaystyle-{\frac{{{\cos{{\left({x}^{{{2}}}\right)}}}}}{{{2}}}}+{C}\)
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