Calculus

# 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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