Lesson 84 — Technique: u-substitution

Calculate $\int (2x+1)^4\, dx$.

Calculate $\int x(x^2 - 3)^5\, dx$.

Calculate $\int \cos^3 x \sin x\, dx$.

Calculate $\int x^2 e^{x^3 + 2}\, dx$.

Calculate $\int \frac{x}{x^2 + 1}\, dx$.

Calculate $\int \cos(3x)\, dx$.

Calculate $\int x e^{-x^2/2}\, dx$.

Calculate $\int \frac{(\ln x)^2}{x}\, dx$.

Calculate $\int \sin^4 x \cos x\, dx$.

Calculate $\int \sqrt{4 - 3x}\, dx$.

Calculate $\int (x+2)(x^2+4x)^3\, dx$.

Calculate $\int \sin(2x)\, dx$.

Calculate $\int_1^2 \frac{x}{\sqrt{x^2+1}}\, dx$.

Calculate $\int_0^1 \frac{e^x}{e^x + 1}\, dx$.

Calculate $\int \cot x\, dx$.

Calculate $\int e^{5x}\, dx$.

Calculate $\int \frac{x^2}{x^3 + 1}\, dx$.

Which substitution $u$ is most suitable for computing $\int 3x^2(x^3+1)^4\, dx$?

When attempting to use substitution $u = g(x)$, you notice that $g'(x)$ is multiplied by a constant different from 1. What should you do?

Calculate $\int_0^{\pi/4} e^{\sin^2 x} \sin x \cos x\, dx$.

Calculate $\int \tan^3 x \sec^2 x\, dx$.

Calculate $\int_1^2 x^2(x^3 + 1)^4\, dx$.

A fixed income fund has a contribution rate of R$ 500 per month with exponential growth: $r(t) = 500 e^{0{,}08t}$ reais per month. Calculate the accumulated balance after 12 months.

Calculate $\int x\cos(x^2)\, dx$.

Calculate $\int \frac{1}{(1+\sqrt{x})\sqrt{x}}\, dx$.

Calculate $\int_0^\pi \cos(\sin x) \cos x\, dx$. (Hint: observe the limits after substitution.)

Calculate $\int \frac{e^x}{e^{2x} + 1}\, dx$.

Try to calculate $\int \sec^3 x\, dx$ by substitution. Identify why simple substitution fails here, and write the answer (which requires integration by parts, Lesson 85).

Prove that $\int f(ax + b)\, dx = \frac{1}{a} F(ax + b) + C$ where $F' = f$ and $a \neq 0$.

A vehicle has acceleration $a(t) = 10e^{-0{,}5t}$ m/s², starting from rest ($v(0) = 0$). Find $v(t)$ using substitution and calculate the velocity at $t = 4$ s.