All Flashcards

What is the Integration by Parts formula?

$\int u , dv = uv - \int v , du$

What is the product rule for differentiation?

$\frac{d}{dx} uv = u cdot v' + v cdot u'$

What are the differences between u-substitution and Integration by Parts?

u-substitution: Reverses the chain rule, simplifies composite functions. | Integration by Parts: Reverses the product rule, simplifies products of functions.

Steps to solve $\int xe^x , dx$ using Integration by Parts?

Choose $u=x$ , $dv=e^x dx$ . 2. Find $du=dx$ , $v=e^x$ . 3. Apply formula: $xe^x - \int e^x dx$ . 4. Evaluate: $xe^x - e^x + C$ .

Steps to solve $\int \ln(x) , dx$ using Integration by Parts?

Rewrite as $\int 1 \cdot \ln(x) , dx$ . 2. Choose $u=\ln(x)$ , $dv=dx$ . 3. Find $du=\frac{1}{x}dx$ , $v=x$ . 4. Apply formula: $x\ln(x) - \int x(\frac{1}{x}) dx$ . 5. Evaluate: $x\ln(x) - x + C$ .

Steps to solve $\int x^2 \cos(x) , dx$ ?

Choose $u=x^2$ , $dv=\cos(x)dx$ . 2. Find $du=2x dx$ , $v=\sin(x)$ . 3. Apply formula: $x^2\sin(x) - \int 2x\sin(x) dx$ . 4. Apply IBP again to $\int 2x\sin(x) dx$ . 5. Final result: $x^2 \sin(x) + 2x \cos(x) - 2\sin(x) + C$ .

Steps to evaluate $\int e^x \cos x , dx$ ?

Apply IBP twice, keeping $e^x$ as part of 'u' or 'dv' consistently. 2. Isolate the original integral using algebraic manipulation. 3. Solve for the original integral.