91Ó°ÊÓ

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. \(\int\left(f^{(p)}(x)\right)^{n} f^{(p+1)}(x) d x,\) where \(p\) is a positive integer, \(n \neq-1\)

Find the area of the region \(R\) bounded by the graph of \(f\) and the \(x\) -axis on the given interval. Graph \(f\) and show the region \(R\) $$f(x)=x^{4}-4 \text { on }[1,4]$$

Evaluate the following integrals in which the function \(f\) is unspecified. Note that \(f^{(p)}\) is the pth derivative of \(f\) and \(f^{p}\) is the pth power of \(f .\) Assume \(f\) and its derivatives are continuous for all real numbers. $$\int 2\left(f^{2}(x)+2 f(x)\right) f(x) f^{\prime}(x) d x$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int_{0}^{1} x \sqrt{x+a} d x ; a>0 \quad(u=\sqrt{x+a} \text { and } u=x+a)$$

Simplify the given expressions. $$\int_{3}^{8} f^{\prime}(t) d t, \text { where } f^{\prime} \text { is continuous on }[3,8]$$

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4}$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int_{0}^{1} x \sqrt[p]{x+a} d x ; a>0 \quad(u=\sqrt[p]{x+a} \text { and } u=x+a)$$

Occasionally, two different substitutions do the job. Use each substitution to evaluate the following integrals. $$\int \sec ^{3} \theta \tan \theta d \theta \quad(u=\cos \theta \text { and } u=\sec \theta)$$

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{\cos x}\left(t^{4}+6\right) d t$$

Simplify the given expressions. $$\frac{d}{d x} \int_{x}^{1} e^{t^{2}} d t$$