hrostentsp6

2021-11-21

Find the indefinite integral $\int \mathrm{cos}h4xdx$

Cherry McCormick

Step 1
To find the indefinite integral.
Step 2
Given information:
$\int \mathrm{cos}h\left(4x\right)dx$
Formula used:
$\int \mathrm{cos}h\left(nv\right)dv=\frac{\mathrm{sin}h\left(nv\right)}{n}+C$
Step 3
Calculation:
$\int \mathrm{cos}h\left(4x\right)dx$
From formula,
$\int \mathrm{cos}h\left(4x\right)dx=\frac{\mathrm{sin}h\left(4x\right)}{4}+C$
This is the indefinite integral.

Stephanie Mann

Step 1: Regroup terms.
$\int h\mathrm{cos}4xdx$
Step 2: Use Constant Factor Rule: $\int cf\left(x\right)dx=c\int f\left(x\right)dx$.
$h\int \mathrm{cos}4xdx$
Step 3: Use Integration by Substitution on $\int \mathrm{cos}4xdx$.
Let u=4x, du=4 dx, then $dx=\frac{1}{4}du$
Step 4: Using u and du above, rewrite $\int \mathrm{cos}4xdx$.
$\int \frac{\mathrm{cos}u}{4}du$
Step 5: Use Constant Factor Rule: $\int cf\left(x\right)dx=c\int f\left(x\right)dx$.
$\frac{1}{4}\int \mathrm{cos}udu$
Step 6: Use Trigonometric Integration: the integral of .
$\frac{\mathrm{sin}u}{4}$
Step 7: Substitute u=4x back into the original integral.
$\frac{\mathrm{sin}4x}{4}$
Step 8: Rewrite the integral with the completed substitution.
$\frac{h\mathrm{sin}4x}{4}$
$\frac{h\mathrm{sin}4x}{4}+C$