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Expand the integral $\int_{1}^{2}\left(4\sqrt[3]{t}+t\left(t^2+1\right)^3\right)dt$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately
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$\int_{1}^{2}4\sqrt[3]{t}dt+\int_{1}^{2} t\left(t^2+1\right)^3dt$
Learn how to solve definite integrals problems step by step online. Integrate the function 4t^1/3+t(t^2+1)^3 from 1 to 2. Expand the integral \int_{1}^{2}\left(4\sqrt[3]{t}+t\left(t^2+1\right)^3\right)dt into 2 integrals using the sum rule for integrals, to then solve each integral separately. We can solve the integral \int_{1}^{2} t\left(t^2+1\right)^3dt by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that t^2+1 it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. Now, in order to rewrite dt in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above. Isolate dt in the previous equation.