Integrals - Problem Solving


Evaluate \[ \int _{ 0 }^{ \frac{\pi}{4} }{ \frac{21 \cos^2 x}{\sin x + \cos x} } dx + \int _{ \frac{\pi}{4} }^{ 0 }{ \frac{21 \sin^2 x}{\sin x + \cos x} } dx. \]

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

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

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

Suppose \(f(x)\) and \(F(x)\) are polynomial functions such that \[F'(x) = f(x) \mbox{ and } F(x)=xf(x)-4x^3+x^2.\] If \(f(0)=6,\) what is \(f(x)?\)


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