Chapter 6: Problem 37
True or False If \(f^{\prime}(x)=g(x),\) then \(\int x^{2} g(x) d x=\) \(x^{2} f(x)-2 \int x f(x) d x .\) Justify your answer.
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In Exercises \(25-28\) , evaluate the integral analytically. Support your answer using NINT. $$\int_{-2}^{3} e^{2 x} \cos 3 x d x$$
In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{(1-x)^{2}}$$
Trigonometric Substitution Suppose \(u=\sin ^{-1} x .\) Then \(\cos u>0\) . (a) Use the substitution \(x=\sin u, d x=\cos u d u\) to show that $$\int \frac{d x}{\sqrt{1-x^{2}}}=\int 1 d u$$ (b) Evaluate \(\int 1 d u\) to show that \(\int \frac{d x}{\sqrt{1-x^{2}}}=\sin ^{-1} x+C\)
In Exercises \(53-66,\) make a \(u\) -substitution and integrate from \(u(a)\) to \(u(b) .\) $$\int_{0}^{1} r \sqrt{1-r^{2}} d r$$
In Exercises \(29-32,\) solve the differential equation. $$\frac{d y}{d x}=x^{2} e^{4 x}$$
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