Find an anti-derivative of the function fx x2 1x3 3x4

Question

The Substitution Rule

Exercise 1- Find an anti-derivative of the function f(x) = (x² + 1)(x³ + 3x)⁴.

Exercise 2- Find an anti-derivative of the function f(x) = sin(ln x)/x.

Exercise 3- Find an anti-derivative of the function f(x) = 2^x/2^x+3.

Exercise 4- Find an anti-derivative of the function f(x) = x/1+x⁴.

Exercise 5- Evaluate the definite integral ₀∫¹xe^{-x^2}dx.

Exercise 6- Evaluate the definite integral _e∫^{e^4}(1/x√ln x)dx.

Exercise 7- If f is continuous and ₀∫⁹f(x) dx = 4, find ₀∫³xf(x²) dx.

Exercise 8- If a and b are positive numbers, show that ₀∫¹x^a(1 - x)^bdx =₀∫¹x^b(1 - x)^a dx.

Exercise 9- If f is continuous on [0, π], use the substitution u = π - x to show that

₀∫^πxf(sin x) dx = π/2 ₀∫^πf(sin x) dx.

Use this to evaluate the integral

₀∫^π(x sin x/1 + cos² x)dx.