Can someone solve these integrals?

Question

a) ∫x^2-3x+2/√x dx

b) ∫e^2 dx

c) ∫(x+1)^3/2x dx

d) ∫dx/4x^2+25

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Answer 1

a) ∫ (x^2-3x+2/√x) dx

= ∫x^2 dx - ∫3xdx + ∫2/√x dx

= ∫x^2 dx -  3∫xdx +2 ∫1/√x dx

The integral of ∫x^n dx = (x^n+1)/(n+1) + c

= (x^2+1) / (2+1) - 3 (x^1+1)/(1+1) + 2 (x^ -1/2+1)/( -1/2+1 )+ c

= (x^3) / 3 - 3 ( x^2 )/2 + 2 ( x^ 1/2)/ (1/2) + c

= (x^3) / 3 - 3 ( x^2 ) /2 + 4 (x^1/2) + c

Answer 2

b) ∫e^2 dx

= ∫e^2 dx

= e^2∫ 1 dx

= e^2 ( x + c)

= x e^2 + e^2 c

Comments

∫e² dx

Integral of e² is e²x.

= e²x + C

Answer 3

d) ∫dx/4x^2+25

= ∫(1/4x^2+25)dx

= ∫(1/(2x)^2+(5)^2)dx

Using integral formula ∫(1/(x)^2+(a)^2)dx = (1/a)tan⁻¹ (x / a) + c

= tan⁻¹ (2x / 5) d/dx (2x / 5)  + c

=(2/5) tan⁻¹ (2x / 5) + c

Comments

Take 25 as common.

Answer 4

a) ∫ (x^2-3x+2/√x) dx

= ∫x^2 dx - ∫3xdx + ∫2/√x dx

= ∫x^2 dx -  3∫xdx +2 ∫1/√x dx

The integral of ∫x^n dx = (x^n+1)/(n+1) + c

= (x^2+1) / (2+1) - 3 (x^1+1)/(1+1) + 2 (x^ -1/2+1)/( -1/2+1 )+ c

= (x^3) / 3 - 3 ( x^2 )/2 + 2 ( x^ 1/2)/ (1/2) + c

= (x^3) / 3 - 3 ( x^2 ) /2 + 4 (x^1/2) + c

=2x^3 - 9x^2  +  24 x^1/2 + 6c

Answer 5

c)

Do long division.