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  • https://math.libretexts.org/Courses/College_of_Southern_Nevada/Calculus_(Hutchinson)/05%3A_Integration/5.06%3A_Substitution_with_Definite_Integrals
    Also, we have the option of replacing the original expression for \(u\) after we find the antiderivative, which means that we do not have to change the limits of integration. \[ \begin{align*}\dfrac{1...Also, we have the option of replacing the original expression for \(u\) after we find the antiderivative, which means that we do not have to change the limits of integration. \[ \begin{align*}\dfrac{1}{2}∫^{π/2}_0\,dθ+\dfrac{1}{2}∫^{π/2}_0 \cos 2θ\,dθ &=\dfrac{1}{2}∫^{π/2}_0\,dθ+\dfrac{1}{2}\left(\dfrac{1}{2}\right)∫^π_0 \cos u \,du \\[4pt] &=\dfrac{θ}{2}\,\bigg|^{θ=π/2}_{θ=0}+\dfrac{1}{4}\sin u\,\bigg|^{u=θ}_{u=0} \\[4pt] &=\left(\dfrac{π}{4}−0\right)+(0−0)=\dfrac{π}{4} \end{align*}\]

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