Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Problem 1
\(1-4\) Verify by differentiation that the formula is correct. $$\int \frac{x}{\sqrt{x^{2}+1}} d x=\sqrt{x^{2}+1}+C$$
Problem 1
Explain exactly what is meant by the statement that "differentiation and integration are inverse processes."
Problem 1
I-6 Evaluate the integral by making the given substitution. \(\int e^{-x} d x, u=-x\)
Problem 2
\(1-4\) Verify by differentiation that the formula is correct. $$\int x \cos x d x=x \sin x+\cos x+c$$
Problem 2
If \(f(x)=x^{2}-2 x, 0 \leqslant x \leqslant 3,\) evaluate the Riemann sum with \(n=6,\) taking the sample points to be right endpoints. What does the Riemann sum represent? Illustrate with a diagram.
Problem 2
1-6 Evaluate the integral by making the given substitution. \(\int x^{3}\left(2+x^{4}\right)^{5} d x, \quad u=2+x^{4}\)
Problem 3
If \(f(x)=e^{x}-2,0 \leq x \leqslant 2,\) find the Riemann sum with \(n=4\) correct to six decimal places, taking the sample points to be midpoints. What does the Riemann sum represent? Illustrate with a diagram.
Problem 3
\(1-4\) Verify by differentiation that the formula is correct. $$\int \cos ^{3} x d x=\sin x-\frac{1}{3} \sin ^{3} x+C$$
Problem 3
(a) Estimate the area under the graph of \(f(x)=\cos x\) from \(x=0\) to \(x=\pi / 2\) using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.
Problem 3
1-6 Evaluate the integral by making the given substitution. \(\int x^{2} \sqrt{x^{3}+1} d x, \quad u=x^{3}+1\)
