Evaluate the integral of (x2 + 4x - 7)(2x + 4) dx using u-substitution.

May 19, 2026Mathematics

ScanSolve AI Answer

May 19, 2026

Answer

$\frac{(x^2 + 4x - 7)^2}{2} + C$

Step 1: Identify a suitable substitution. Let $u = x^2 + 4x - 7$ .

Step 2: Differentiate $u$ with respect to $x$ to find $du$ . $\frac{du}{dx} = \frac{d}{dx}(x^2 + 4x - 7) = 2x + 4$ So, $du = (2x + 4)\,dx$ .

Step 3: Substitute $u$ and $du$ into the integral. $\int (x^2 + 4x - 7)(2x + 4)\,dx = \int u\,du$

Step 4: Integrate with respect to $u$ . $\int u\,du = \frac{u^{1+1}}{1+1} + C = \frac{u^2}{2} + C$

Step 5: Substitute back $u = x^2 + 4x - 7$ into the result. $\frac{(x^2 + 4x - 7)^2}{2} + C$

The final answer is $\boxed{\frac{(x^2 + 4x - 7)^2}{2} + C}$ .

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Quick Answer

(du)/(dx) = (d)/(dx)(x^2 + 4x - 7) = 2x + 4 So, du = (2x + 4)\,dx.
(x^2 + 4x - 7)(2x + 4)\,dx = u\,du Step 4: Integrate with respect to u.
u\,du = u^1+11+1 + C = (u^2)/(2) + C Step 5: Substitute back u = x^2 + 4x - 7 into the result.
((x^2 + 4x - 7)^2)/(2) + C The final answer is ((x^2 + 4x - 7)^2)/(2) + C.

Asked on May 19, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on May 19, 2026

Step 1: Identify a suitable substitution. Let u = x^2 + 4x - 7. Step 2: Differentiate u with respect to x to find du. (du)/(dx) = (d)/(dx)(x^2 + 4x - 7) = 2x + 4 So, du = (2x + 4)\,dx. Step 3: Substitute u and du into the integral. (x^2 + 4x - 7)(2x + 4)\,dx = u\,du Step 4: Integrate with respect to u. u\,du = u^1+11+1 + C = (u^2)/(2) + C Step 5: Substitute back u = x^2 + 4x - 7 into the result. ((x^2 + 4x - 7)^2)/(2) + C The final answer is ((x^2 + 4x - 7)^2)/(2) + C. Send me the next one 📸