What is the integral of sqrt(x+1)cos(sqrt(x+1)^3+2) ?

The integral of sqrt(x+1)cos(sqrt(x+1)^3+2) is 2/3 sin((x+1)^{3/2}+2)+C

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integral of sqrt(x+1)cos(sqrt(x+1)^3+2)

\int x + 1 cos (x + 1^{3} + 2) d x

\frac{2}{3} sin ((x + 1)^{\frac{3}{2}} + 2) + C

Solution steps

\int x + 1 cos ((x + 1)^{3} + 2) d x

Apply u-substitution

: \int u cos (u^{\frac{3}{2}} + 2) d u

= \int u cos (u^{\frac{3}{2}} + 2) d u

Apply u-substitution

: \int 2 v^{2} cos (v^{3} + 2) d v

= \int 2 v^{2} cos (v^{3} + 2) d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int v^{2} cos (v^{3} + 2) d v

Apply u-substitution

: \int \frac{cos ( w )}{3} d w

= 2 \cdot \int \frac{cos ( w )}{3} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \frac{1}{3} \cdot \int cos (w) d w

Use the common integral:

\int cos (w) d w = sin (w)

= 2 \cdot \frac{1}{3} sin (w)

Substitute back

= 2 \cdot \frac{1}{3} sin ((x + 1)^{3} + 2)

Simplify

2 \cdot \frac{1}{3} sin ((x + 1)^{3} + 2) : \frac{2}{3} sin ((x + 1)^{\frac{3}{2}} + 2)

= \frac{2}{3} sin ((x + 1)^{\frac{3}{2}} + 2)

Add a constant to the solution

= \frac{2}{3} sin ((x + 1)^{\frac{3}{2}} + 2) + C

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What is the integral of sqrt(x+1)cos(sqrt(x+1)^3+2) ?
The integral of sqrt(x+1)cos(sqrt(x+1)^3+2) is 2/3 sin((x+1)^{3/2}+2)+C