1b Integrate the following expressions.
∫(3𝑥3−4𝑥2+5𝑥−3)𝑑𝑥
∫(3𝑐𝑜𝑠2𝜃+5𝑐𝑜𝑠4𝜃)𝑑𝜃
1) ∫(3𝑥3−4𝑥2+5𝑥−3)𝑑𝑥=3∫x3dx−4∫x2dx+5∫xdx−3∫1 dx∫(3𝑥^3−4𝑥^2+5𝑥−3)𝑑𝑥=3\int x^3dx-4\int x^2dx+5 \int xdx-3\int 1 \ dx∫(3x3−4x2+5x−3)dx=3∫x3dx−4∫x2dx+5∫xdx−3∫1 dx
⇒34x4−43x3+52x2−3x+C\Rightarrow \dfrac{3}{4}x^4-\dfrac{4}{3}x^3 +\dfrac{5}{2}x^2-3x+C⇒43x4−34x3+25x2−3x+C
2) ∫(3cos2θ+5cos4θ)dθ=3∫cos2θdθ+5∫cos4θdθ\int(3cos2\theta+5cos4\theta)d\theta=3\int cos2\theta d\theta+5\int cos4\theta d\theta∫(3cos2θ+5cos4θ)dθ=3∫cos2θdθ+5∫cos4θdθ
⇒32sin2θ+54sin4θ+C\Rightarrow \dfrac{3}{2}sin2\theta+\dfrac{5}{4}sin4\theta+C⇒23sin2θ+45sin4θ+C
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