The antiderivative of arctan(4t)dt by integration of parts.
| = integration symbol
| u dv = uv - | v du
| arctan(4t) dt
u = arctan(4t)
du = 4/(16t^2 + 1) dt
dv = dt
v = t
It should look like this,
= t arctan(4t) - | 4t/(16t^2 + 1) dt
After you set that up, substitute
w = 16t^2 + 1
dw = 32t dt
1/32 dw = t dt
Then, the integration should look like this,
= t arctan(4t) - 1/8 | 1/w dw
Make sure I have the correct variables, t/dt as opposed to x/dx (except where I substitute w and dw for t/dt)
Post back if you get stuck. This is a fairly easy integration (except for the substitutions).
To find the antiderivative of the function arctan(4t)dt using integration by parts, we need to follow these steps:
Step 1: Identify the parts of the function
In integration by parts, we split the function into two parts: u and dv. We then differentiate u and integrate dv. For this problem, we can choose u = arctan(4t) and dv = dt.
Step 2: Find du and v
To find du, we differentiate u with respect to t. In this case, du/dt = 4/(1+16t^2). To find v, we integrate dv with respect to t. Since dv = dt, the integral of dv is simply v = t.
Step 3: Apply the integration by parts formula
The integration by parts formula states that ∫ u dv = uv - ∫ v du. Using this formula, we can find the antiderivative of the function.
∫ arctan(4t)dt = u*v - ∫ v*du
∫ arctan(4t)dt = arctan(4t)*t - ∫ t * (4/(1+16t^2))dt
Step 4: Simplify and integrate
Now, we can simplify the integral and compute the antiderivative.
∫ arctan(4t)dt = arctan(4t)*t - 4∫ (t/(1+16t^2))dt
To integrate the second term on the right side, we can use a substitution. Let u = 1 + 16t^2. Then, du = 32tdt.
∫ arctan(4t)dt = arctan(4t)*t - 4[(1/32) ∫ (t/((1+16t^2)) * 32tdt]
∫ arctan(4t)dt = arctan(4t)*t - (1/8) ∫ (t/((1+16t^2)) du
Now, we can substitute u back into the equation.
∫ arctan(4t)dt = arctan(4t)*t - (1/8) ∫ (1/u) du
∫ arctan(4t)dt = arctan(4t)*t - (1/8) ln|u| + C
Step 5: Simplify the expression
Finally, we can simplify the expression by substituting back u into the equation.
∫ arctan(4t)dt = arctan(4t)*t - (1/8) ln|1+16t^2| + C
Therefore, the antiderivative of arctan(4t)dt using integration by parts is arctan(4t)*t - (1/8) ln|1+16t^2| + C, where C is the constant of integration.