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Use the Taylor series for rewrite the function $e^x$ as an approximation: $\displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$, with $a=0$. Here we will use only the first four terms of the serie to approximate the function
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$\int e^x\sin\left(x\right)^2dx$
Learn how to solve problems step by step online. Find the integral int(e^xsin(x)^2)dx. Use the Taylor series for rewrite the function e^x as an approximation: \displaystyle f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n, with a=0. Here we will use only the first four terms of the serie to approximate the function. Any expression divided by one (1) is equal to that same expression. Rewrite the integrand \left(1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}\right)\sin\left(x\right)^2 in expanded form. Expand the integral \int\left(\sin\left(x\right)^2+x\sin\left(x\right)^2+\frac{1}{2}x^{2}\sin\left(x\right)^2+\frac{1}{6}x^{3}\sin\left(x\right)^2\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately.