Find the integral of these two equations.?

Question

Use u-substitution and include steps please. I have only been able to find examples which have the complicated function on the bottom rather than on top. please help!

1) (1-sin^2x)/(cosx)dx
2) (x^4+1)/(x)dx

Answer 1

1). ʃ[(1 - sin²x)/(cos(x))]dx

Pythagorean Identities: sin²θ + cos²θ = 1 ⇒ cos²θ = 1 - sin²θ

ʃ[cos²x/cos(x)]dx

ʃcos(x)dx

Integrals of trigonometric function: ʃcos(θ)dθ = sinθ + c

sin(x) + c.

 

2). ʃ[(x⁴ + 1)/x]dx

ʃ[x⁴/x + 1/x]dx

ʃ[x³ +1/x ]dx

ʃ[x³]dx + ʃ[1/x ]dx

Integrals of rational function formula: ʃxⁿ = [xⁿ⁺¹]/[n+1] + c and ʃ[1/x]dx = log(x) + c.

[x⁴/4] + log(x) + c.