integrate 3/25+4x^3....use 2x=5tanΦ as a substitution
To integrate the expression (3/25 + 4x^3) using the substitution 2x = 5tanΦ, you need to follow a few steps.
Step 1: Rewrite the equation in terms of Φ.
By substituting 2x = 5tanΦ, we solve for x as x = (5/2)tanΦ. Now, let's rewrite the expression in terms of Φ.
3/25 + 4x^3
= 3/25 + 4((5/2)tanΦ)^3
= 3/25 + 4(125/8)tan^3Φ
= 3/25 + 500/8 * tan^3Φ
= 3/25 + 250/4 * tan^3Φ
= 3/25 + 250/4 * tan^2Φ * tanΦ
Step 2: Find the derivative of 2x = 5tanΦ.
To substitute the dx in the integral, we also need to find dx in terms of dΦ.
Substituting 2x = 5tanΦ:
Differentiating both sides, we get:
2 dx = 5 sec^2Φ dΦ
dx = (5/2) sec^2Φ dΦ
Step 3: Substitute the expression and dx in the integral.
Now, the integral becomes:
∫ [(3/25) + (250/4)tan^2Φ * tanΦ] * (5/2)sec^2Φ dΦ
Step 4: Simplify the expression.
Since tan^2Φ = sec^2Φ - 1, we can rewrite the integral as:
∫ [(3/25) + (250/4)(sec^2Φ - 1)tanΦ] * (5/2)sec^2Φ dΦ
Expanding further:
∫ [(3/25) + (250/4)sec^2ΦtanΦ - (250/4)tanΦ] * (5/2)sec^2Φ dΦ
Step 5: Distribute and combine like terms.
∫ (15/50 + (1250/8)sec^2ΦtanΦ - (1250/8)tanΦ) sec^2Φ dΦ
Step 6: Simplify the expression and integrate.
∫ [(3/10) + (1250/8)sec^2Φtan^2Φ - (1250/8)tanΦsec^2Φ] dΦ
Now, you can perform integration on each term. Please note that the integral of sec^2Φ is tanΦ, and the integral of tanΦ is -ln|secΦ| + C, where C is the constant of integration.
After integrating each term, you will get the final answer in terms of Φ.