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Evaluate ∫0π421cos2xsinx+cosxdx+∫π4021sin2xsinx+cosxdx. \int _{ 0 }^{ \frac{\pi}{4} }{ \frac{21 \cos^2 x}{\sin x + \cos x} } dx + \int _{ \frac{\pi}{4} }^{ 0 }{ \frac{21 \sin^2 x}{\sin x + \cos x} } dx. ∫04πsinx+cosx21cos2xdx+∫4π0sinx+cosx21sin2xdx.
If f(x)=∫(x−1)2xdxf(x)=\int \frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}} dxf(x)=∫x(x−1)2dx and f(1)=53,\displaystyle f(1)=\frac{5}{3},f(1)=35, what is the value of 3f(16)?3f(16)?3f(16)?
Let f(x)=∫0x7+6t2dt.\displaystyle f(x)=\int_{0}^{x}{\sqrt{7+6{t}^{2}}dt.}f(x)=∫0x7+6t2dt. Then what are the real roots of the equation x2−df(x)dx=0?{x}^{2}-\frac{df(x)}{dx}=0?x2−dxdf(x)=0?
If f(m)f(m)f(m) is the number of the points crossed by the curve y=x2−mx+m y=x^2-mx+my=x2−mx+m and the xxx-axis, what is ∫712f(m) dm?\displaystyle \int_{7}^{12} f(m) \,dm ?∫712f(m)dm?
Suppose f(x)f(x)f(x) and F(x)F(x)F(x) are polynomial functions such that F′(x)=f(x) and F(x)=xf(x)−4x3+x2.F'(x) = f(x) \mbox{ and } F(x)=xf(x)-4x^3+x^2.F′(x)=f(x) and F(x)=xf(x)−4x3+x2. If f(0)=6,f(0)=6,f(0)=6, what is f(x)?f(x)?f(x)?
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