## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\frac{\pi}{2}}\left(1-5 x^{2}\right) \sin x \, dx
$$
## Solution

$$
\int_{0}^{\frac{\pi}{2}}\left(1-5 x^{2}\right) \sin x d x=
$$

Let:

$$
\begin{aligned}
& u=1-5 x^{2} ; d u=-10 x \cdot d x \\
& d v=\sin x d x ; v=-\cos x
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =\left.\left(1-5 x^{2}\right) \cdot(-\cos x)\right|_{0} ^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}}(-\cos x) \cdot(-10 x) \cdot d x= \\
& =\left(1-5 \cdot\left(\frac{\pi}{2}\right)^{2}\right) \cdot\left(-\cos \frac{\pi}{2}\right)-\left(1-5 \cdot 0^{2}\right) \cdot(-\cos 0)-10 \cdot \int_{0}^{\frac{\pi}{2}} \cos x \cdot x \cdot d x= \\
& =\left(1-5 \cdot\left(\frac{\pi}{2}\right)^{2}\right) \cdot 0-1 \cdot(-1)-10 \cdot \int_{0}^{\frac{\pi}{2}} \cos x \cdot x \cdot d x= \\
& =1-10 \cdot \int_{0}^{\frac{\pi}{2}} \cos x \cdot x \cdot d x=
\end{aligned}
$$

Let:

$$
\begin{aligned}
& u=x ; d u=d x \\
& d v=\cos x d x ; v=\sin x
\end{aligned}
$$

Using the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =1-10 \cdot\left(\left.x \cdot \sin x\right|_{0} ^{\frac{\pi}{2}}-\int_{0}^{\frac{\pi}{2}} \sin x \cdot d x\right)= \\
& =1-10 \cdot\left(\frac{\pi}{2} \cdot \sin \frac{\pi}{2}-0 \cdot \sin 0-\int_{0}^{\frac{\pi}{2}} \sin x d x\right)= \\
& =1-10 \cdot\left(\frac{\pi}{2} \cdot 1-0-\left.(-\cos x)\right|_{0} ^{\frac{\pi}{2}}\right)= \\
& =1-10 \cdot\left(\frac{\pi}{2}+\cos \frac{\pi}{2}-\cos 0\right)=1-10 \cdot\left(\frac{\pi}{2}+0-1\right)= \\
& =1-5 \pi+10=11-5 \pi
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A\%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-19$ » Categories: Kuznetsov's Problem Book Integrals Problem $2 \mid$ Integrals

- Last edited: 19:37, 27 February 2009.
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## Problem Kuznetsov Integrals 2-20

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