 # 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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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.
###### Have a similar question? Befory
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.
$$\displaystyle-{\frac{{{\cos{{\left({x}^{{{2}}}\right)}}}}}{{{2}}}}$$
$$\displaystyle-{\frac{{{\cos{{\left({x}^{{{2}}}\right)}}}}}{{{2}}}}+{C}$$