Lesson 85 — Integration by Parts

Compute $\int x e^x\, dx$.

Compute $\int x \sin x\, dx$.

Compute $\int x \cos x\, dx$.

Compute $\int \ln x\, dx$.

Compute $\int x^2 e^x\, dx$. Apply parts twice or use the tabular method.

Compute $\int x^2 \sin x\, dx$.

Compute $\int x^2 \cos x\, dx$.

Compute $\int x \ln x\, dx$.

Compute $\int x^2 \ln x\, dx$.

Compute $\int \arctan x\, dx$.

Compute $\int \arcsin x\, dx$.

Compute $\int x e^{2x}\, dx$.

Compute $\int x \cdot 2^x\, dx$.

Compute $\int x e^{-x}\, dx$.

Compute $\int (\ln x)^2\, dx$. Apply parts twice.

Compute $\int e^x \cos x\, dx$.

Compute $\int e^x \sin x\, dx$.

Compute $\int e^{2x} \cos(3x)\, dx$.

Compute $\int e^{-x} \sin(2x)\, dx$.

Compute $\int \cos x \ln(\sin x)\, dx$.

Compute $\int \sec^3 x\, dx$.

Compute $\int \csc^3 x\, dx$.

Compute $\int_0^1 x e^x\, dx$.

Compute $\int_0^\pi x \sin x\, dx$.

Compute $\int_1^e \ln x\, dx$.

Compute $\int_0^{\pi/2} x \cos x\, dx$.

Compute $\int_0^1 \arctan x\, dx$.

Compute $\int_1^2 x^2 \ln x\, dx$.

Compute $\int_0^1 x e^{-x}\, dx$.

Compute $\int_0^{\pi/2} e^x \sin x\, dx$.

Work done by force $F(x) = x e^{-x}$ N along $[0, 1]$ m. Compute $W = \int_0^1 F(x)\, dx$.

Electric charge accumulated with current $i(t) = t\cos(\omega t)$. Compute $Q = \int_0^{2\pi/\omega} i(t)\, dt$.

Gamma function: compute $\Gamma(2) = \int_0^\infty t e^{-t}\, dt$ and show that the result is $1$.

Expectation of exponential variable: compute $E[X] = \int_0^\infty x \lambda e^{-\lambda x}\, dx$ and show that the result is $1/\lambda$.

Variance of exponential: compute $E[X^2] = \int_0^\infty x^2 \lambda e^{-\lambda x}\, dx$ and determine $\text{Var}(X) = E[X^2] - (E[X])^2$.

Fourier coefficient $a_n = (1/\pi)\int_{-\pi}^\pi x\cos(nx)\, dx$. Determine $a_n$ for all integers $n \geq 1$.

Laplace transform of $t$: show that $\mathcal{L}\{t\}(s) = \int_0^\infty t e^{-st}\, dt = 1/s^2$ for $s > 0$.

Present value of growing income $C(t) = t$ (in thousands of reais/year) with continuous discount rate $r$ over $[0, T]$. Compute $VP = \int_0^T t e^{-rt}\, dt$.

Compute $\int x^3 e^{-x^2}\, dx$. Hint: substitute $u = x^2$ first, then apply parts.

Compute $\int e^{\sqrt{x}}\, dx$. Hint: substitute $u = \sqrt{x}$, then apply parts.

Show by induction that $\int_0^\infty x^n e^{-x}\, dx = n!$ for all non-negative integer $n$, using integration by parts in the inductive step.

Let $I_n = \int x^n e^x\, dx$. Show that $I_n = x^n e^x - n I_{n-1}$ and use the formula to compute $I_3$.

Derive the reduction formula $\int \sin^n x\, dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int \sin^{n-2}x\, dx$ and apply it to compute $\int \sin^4 x\, dx$.

Proof. Prove the formula $\int u\, dv = uv - \int v\, du$ from the product rule for derivatives and the Fundamental Theorem of Calculus.

Informal Proof. Justify why LIATE is an effective heuristic: analyze how each type of function behaves upon differentiation and explain why Log and Inv-trig are the top candidates for $u$.