Find the Antiderivative f(x)=3x^5+x-4

Problem

(∫_^)(3*x5+x−4*d(x))

Solution

1. Apply the sum rule for integration, which allows the integral of a sum to be written as the sum of the integrals of each term.

(∫_^)(3*x5+x−4*d(x))=(∫_^)(3*x5*d(x))+(∫_^)(x*d(x))−(∫_^)(4*d(x))

2. Apply the power rule for integration, (∫_^)(xn*d(x))=(x(n+1))/(n+1) to each term.

(∫_^)(3*x5*d(x))=3⋅(x6)/6

(∫_^)(x1*d(x))=(x2)/2

(∫_^)(4*d(x))=4*x

3. Simplify the coefficients of the resulting terms.

3⋅(x6)/6=1/2*x6

4. Combine the terms and add the constant of integration C

1/2*x6+1/2*x2−4*x+C

Final Answer

(∫_^)(3*x5+x−4*d(x))=1/2*x6+1/2*x2−4*x+C

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