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${:[int(3x+4)cos xdx],[intx^(5)sqrt(x^(3)+1)dx]:}$

$\begin{array}{l}\int(3 x+4) \cos x d x \\ \int x^{5} \sqrt{x^{3}+1} d x\end{array}$

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Evaluate definite integrals using the chain rule

Full solution

Q.

\begin{array}{l}\int(3 x+4) \cos x d x \\ \int x^{5} \sqrt{x^{3}+1} d x\end{array}

Solve Integral using Integration by Parts: Step $1$ : Solve the integral of $(3x + 4) \cos x \, dx$ . Using integration by parts, let $u = 3x + 4$ and $dv = \cos x \, dx$ . Then, $du = 3 \, dx$ and $v = \sin x$ . Apply the integration by parts formula: $\int u\,dv = uv - \int v\,du$ . $\int(3x + 4) \cos x \, dx = (3x + 4) \sin x - \int 3 \sin x \, dx$ .
Continue Solving Integral: Step $2$ : Continue solving the integral of $3 \sin x \, dx$ .
$\int 3 \sin x \, dx = -3 \cos x + C$ .
Substitute back into the integration by parts formula:
$(3x + 4) \sin x - (-3 \cos x + C) = (3x + 4) \sin x + 3 \cos x + C$ .
Solve Integral using Substitution: Step $3$ : Solve the integral of $x^5 \sqrt{x^3 + 1} \, dx$ . Let $u = x^3 + 1$ , then $du = 3x^2 \, dx$ . Rewrite $x^5$ as $x^3 \times x^2$ to use substitution: $x^5 = (u - 1) \times x^2$ . But, we need $x^2 \, dx$ for substitution, which isn't directly available.

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