Integrals - Problem Solving


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.

If f(x)=(x1)2xdxf(x)=\int \frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}} dx and f(1)=53,\displaystyle f(1)=\frac{5}{3}, what is the value of 3f(16)?3f(16)?

Let f(x)=0x7+6t2dt.\displaystyle f(x)=\int_{0}^{x}{\sqrt{7+6{t}^{2}}dt.} Then what are the real roots of the equation x2df(x)dx=0?{x}^{2}-\frac{df(x)}{dx}=0?

If f(m)f(m) is the number of the points crossed by the curve y=x2mx+m y=x^2-mx+m and the xx-axis, what is 712f(m)dm?\displaystyle \int_{7}^{12} f(m) \,dm ?

Suppose f(x)f(x) and 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. If f(0)=6,f(0)=6, what is f(x)?f(x)?


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