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Substitution by parts: blem 10
t)
ate the indefinite integral.
)

xsin^(2)(5x)dx=◻+C.
Hint: Integrate by parts with
u=x.
gauss.vaniercollege.qc.ca
gauss.vaniercollege.qc.ca

Substitution by parts: blem $10$ $\newline$ t) $\newline$ ate the indefinite integral. $\newline$ ) $\newline$ $x \sin ^{2}(5 x) d x=\square+C$ . $\newline$ Hint: Integrate by parts with $u=x$ . $\newline$ gauss.vaniercollege.qc.ca $\newline$ gauss.vaniercollege.qc.ca

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Find indefinite integrals using the substitution

Full solution

Q. Substitution by parts: blem

10

\newline

t)

\newline

ate the indefinite integral.

\newline

)

\newline

x \sin ^{2}(5 x) d x=\square+C

.

\newline

Hint: Integrate by parts with

u=x

.

\newline

gauss.vaniercollege.qc.ca

\newline

gauss.vaniercollege.qc.ca

Identify $u$ and $dv$ : Let's use integration by parts where $u = x$ and $dv = \sin^{2}(5x)\,dx$ .
Find $du$ and $v$ : First, we need to find $du$ and $v$ . $du = dx$ and for $v$ , we need to integrate $\sin^{2}(5x)\,dx$ .
Integrate $\sin^2(5x)$ : To integrate $\sin^{2}(5x)$ , use the power reduction formula: $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}$ . $\newline$ So, $\int \sin^{2}(5x)\,dx = \int \frac{1 - \cos(10x)}{2} \,dx$ .
Apply power reduction formula: Now integrate: $\int(\frac{1}{2})dx - \int(\frac{\cos(10x)}{2})dx = (\frac{x}{2}) - (\frac{\sin(10x)}{20}) + C$ . So, $v = (\frac{x}{2}) - (\frac{\sin(10x)}{20})$ .
Calculate $v$ : Now apply integration by parts: $\int u\,dv = uv - \int v\,du$ . Plug in $u$ , $du$ , and $v$ : \int x\sin^{\(2\)}(\(5x)\,dx = x\left(\frac{x}{ $2$ } - \frac{\sin( $10$ x)}{ $20$ }\right) - \int\left(\frac{x}{ $2$ } - \frac{\sin( $10$ x)}{ $20$ }\right)dx.
Apply integration by parts: Simplify the integral: x\left(\frac{x}{\(2\)} - \frac{\sin(\(10\)x)}{\(20\)}\right) - \int\left(\frac{x}{\(2\)}\right)dx + \int\left(\frac{\sin(\(10\)x)}{\(20\)}\right)dx.
Simplify the integral: Integrate each part: \(\frac{x^2}{2} - $\frac{x\sin(10x)}{20}$ - $\frac{x^2}{4}$ + $\frac{\cos(10x)}{200}$ + $C$ .
Integrate each part: Combine like terms: $(\frac{x^2}{4}) - (\frac{x\sin(10x)}{20}) + (\frac{\cos(10x)}{200}) + C$ .
Combine like terms: Oops, made a mistake in the integration by parts formula. It should be $uv - \int v\,du$ , not $uv - \int u\,dv$ . Let's correct that.

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