Integrate the function $\left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right)$

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$\int\left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right)dx$

Learn how to solve integral calculus problems step by step online. Integrate the function (a^(x+1)-2b^(x-1))(2b^(x-1)+a^(x+1)). Find the integral. Rewrite the integrand \left(a^{\left(x+1\right)}-2b^{\left(x-1\right)}\right)\left(2b^{\left(x-1\right)}+a^{\left(x+1\right)}\right) in expanded form. Expand the integral \int\left(a^{\left(2x+2\right)}-4b^{\left(2x-2\right)}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. We can solve the integral \int a^{\left(2x+2\right)}dx 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 2x+2 it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part.